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363 terms

log(b)1

0

log(b)b

1

log(b)b^x

x

b^log(b)x

x

log1

0

log10

1

log10^x

x

10^logx

x

ln1

0

lne

1

lne^x

x

e^lnx

x

The Product Rule: log(b)(MN)

log(b)N+log(b)M

The Quotient Rule: log(b)(M/N)

log(b)N-log(b)M

The Power Rule: log(b)M^p

plog(b)M

Change of Base (common logs): log(b)M

logM/logb

Change of Base (natural logs): log(b)M

lnM/lnb

The exponential function with base b

f(x)=b^x

The natural exponential function

f(x)=e^x

Formula for interest compounded n times per year

A=P(1+r/n)^(nt)

A= balance

P= prinicpal

t= time in years

n= number of times compounded

r= interest rate (in decimal form)

A= balance

P= prinicpal

t= time in years

n= number of times compounded

r= interest rate (in decimal form)

Formula for interest compounded continuously

A=Pe^(rt)

A= balance

P= prinicpal

t= time in years

r= interest rate (in decimal form)

A= balance

P= prinicpal

t= time in years

r= interest rate (in decimal form)

Express in logarithmic form: y=b^x

log(b)x=y

Express in exponential form: log(b)x=y

y=b^x

e

An irrational number called the natural base

e is about 2.7183

e is about 2.7183

The common log logx means:

log(10)x

The natural log lnx means:

log(e)x

Describe the transformation f(x)=b^(x)+c

Shifts the graph c units upward

Describe the transformation f(x)=b^(x)-c

Shifts the graph c units downward

Describe the transformation f(x)=b^(x+c)

Shifts the graph c units left

Describe the transformation f(x)=b^(x-c)

Shifts the graph c units right

Describe the transformation f(x)=-b^x

Reflects the graph about the x-axis

Describe the transformation f(x)=b^(-x)

Reflects the graph about the y-axis

Describe the transformation f(x)=cb^x

Vertical Stretch c>1

Vertial Shrink 0<c<1

Multiply the y-coordinate of each point by "c"

Vertial Shrink 0<c<1

Multiply the y-coordinate of each point by "c"

log(b)N+log(b)M

log(b)(MN)

log(b)N-log(b)M

log(b)(M/N)

plog(b)M

log(b)M^p

Logarithm

the exponent required to produce a given number, this of a positive number y to the base b is defined as follows: If y=b^x, then log b y=x

Asymptote

a line that a graph approaches but never touches

Exponential Function

A function in which the exponent is a variable

Base

watever is being raised to a power

Exponential Growth

When a graph or function changes by increasing amounts

Exponential Decay

When a graph or function changes by decreasing amounts

Logarithm

b ^x=y log y=x

b

b

Natural Log

e^x=y lny=x

General Form

log-alog (x-h) +k

b

b

General Form of a Natural Log

aln (x-h) +k

General Form of Exponential

y=a times b^ x-h +k

Simple Interest

Total = Principal + Principal x interest rate x time in years

Interest for one year

Total = Principal x (1 + interest rate)^time in years

Interest for 5 years compounded monthly

Total = Principal x (1 + interest rate/12)^5 x 12

Interest compounded continuously

Total = Principal x e^interest rate x time in years

Population

Find the percent increase or decrease, add or subtract from one, find a when x=0 and write equation - or set up two equations like y=ab^x

Continuously growing populations

N = N0e^kt

Half Life

Find constant (.5N = N0e^k x half-life) Substitute real left over percentage for the .5 to find the years.

Waves

Find scientific notation for large number and take log of number in order to place on number line

Between each level on a logarithmic scale

it increases by a power of 10 (2 levels = 100x)

Finding time difference

take number and raise 10 to that number, divide/subtract it from the other number 10 is raised to - use new number to determine how many times

pH

take negative log of scientific notation

Hydrogen Ion Concentration

take pH and make it the negative exponent of 10, enter into calculator to find scientific notation

Times difference between pH's

take -log of all scientific notations raise 10 to the power of pH's and subtract lower from higher. Raise 10 to the resulting number to find times.

Function

a set of ordered pairs in which no first element repeats

One-to-one Function

a function in which no second element repeats

Exponential Function

a function of the form f(x)=ab×, where the coefficient a≠0, the base b>0 and b≠1

Inverse Function: First Definition

two function f(x) and g(x) are inverse functions of each other if both are one-to-one functions and for every element in their domain f[g(x)]=g[f(x)]=x. The symbol for the inverse function is f(x) is f^-1(x)

Inverse Function: Second Definition

for any one-to-one function f(x), its inverse, f^-1(x), is defined by the following statement: (a,b)is contained in f(x) if and only if (b,a) is contained in f^-1(x)

Inverse Function: Third Definition

The one-to-one functions f(x) and g(x) are each other's inverses if and only if their graphs are symmetric with respect to the diagonal line f(x)=x

y=log(base)bX is by definition equal to

b^y=X where b>0 and b≠1

Common Logarithm

log(base)10X or log X. THEREFORE y=log X is by def'n 10^y=X

logₐa

1

logₐ1

0

logₐ(1/a)

-1

logₐm+logₐn

logₐmn

logₐm-logₐn

logₐ(m/n)

logₐxⁿ

nlogₐx

logₐ(1/x)

-logₐx

aⁿ=b

logₐb=n

logₓa

logₑa/logₑx

logₐaⁿ

n

exponential function

Includes a constant raised to a variable power, f(x) = b^x. The base b must be positive but cannot equal 1

exponential growth

the graph of an exponential function with a base greater than 1

continuous

a smooth curve; there are no gaps in the curve for the domain

horizontal asymptote

a horizontal line that the curve approaches but never reaches

half-life

a fixed period of time in which something repeatedly decreases by half

compounded annually

Interest that builds on itself at 12 month intervals

equivalent equations

All values for x and y that make one equation true also make the other one true ( b^x = b^y if and only if x=y)

one-to-one function

A function that matches each output with one input

logarithmic function

the inverse of an exponential function

inverse function

A function that reverses the effect of another function

product rule for logarithms

states that the logarithm of a product of numbers equals the sum of the logarithms of the factors (log2 4*8 = ?)

quotient rule for logarithms

states that the logarithm of the quotient of two numbers equals the difference of the logarithms of those numbers (log3 81/3 = ?)

power rule of logarithms

states that the logarithm of a power of M can be calculated as the product of the exponent and the logarithm of M (log2 8^16 = ?)

change-of-base formula

State log16 32 as an expression using 2 base logarithms

common logarithm

logarithms with base 10

sound intensity

a measure of how much power sound transmits

sound level

measured in units called decibels (dB); provides a scale that relates how humans perceive sound to a physical measure of its power

irrational constant

The number 'e'. A number that repeats without pattern

natural logarithm

A logarithm with base 'e'

Napierian logarithm

AKA natural logarithm, named after John Napier, a Scottish theologian and mathematician who discovered logarithms

natural base exponential function

a function of form f(x) = ae^rx

continuously compounded interest

interest that builds on itself at every moment f(t) = Pe^rt

Newton's law of cooling

According to this law, the falling temperature obeys an exponential equation (y = ae^cx + T0, where T0 is the temperature surrounding the cooling object , x is the amount of time, and y is the current temperature)

1

log 10

1

ln e

1/2

2⁻¹

1/9

3⁻²

2

log₃9

2

½⁻¹

2

log₆36

5

ln e⁵

Undefined

ln -2

4

64¹/³

1/125

5⁻³

1/4

8⁻²/³

-1

log₂½

-1

log₄¼

0

log₆1

0

log₂/₃1

undefined

ln 0

27

3³

1/10

100⁻¹/²

2

32¹/⁵

1/5

log₃₂2

1/3

log₈2

-1/3

log₁/₈2

0

ln 1

undefined

log 0

log₂mn

log₂m + log₂n

log₇(m/n)

log₇m - log₇n

log₈x⁷

7log₈x

log₂2

1

log₅1

0

-log₂x

log₂(1/x)

Change the base of log₇x to natural log

ln(x)/ln(7)

percent

a ratio whose denominator is 100

means per hundred

means per hundred

percent change in words

how much a quantity increases or decreases with respect to the original amount

percent change equation

p/100=amount of change/original amount

principal

the original amount of money invested or borrowed

interest

the amount earned or paid for the use of the principal

simple interest

interest is paid only on the principal and only once

compound interest

interest is payed on the principal and previous interest payments

usually compounded at regular intervals

usually compounded at regular intervals

balance

the amount of money in an account at a given time

logarithm (honors)

solves equations for the exponent

an inverse operation to exponents

an inverse operation to exponents

basic logarithms (honors)

y=b^x

logby=x

read as log base b of x

invented in 1614 by John Napier

logby=x

read as log base b of x

invented in 1614 by John Napier

change of base property (honors)

logbm=logm/logb

logbm=lnm/lnb

logbm=lnm/lnb

Exponential Function

y=ab^x

Growth Factor

b>1

Decay Factor

b<1

Rate of Change

the percent increase or decrease of an exponential function, (the distance b is away from zero)

Horizontal Asymptote

Imaginary line that acts as the lowest boundary of an exponential function, since it gets very close to zero, but not exactly zero

"a" in the exponential function equation

starting value

"b" in the exponential function equation

rate of change

Compound Interest Formula

A=P(1+(r/n))^(n/t)

"P" in the compound interest formula

starting value

"r" in the compound interest formula

interest rate

"n" in the compound interest formula

number of times compounded per year

"t" in the compound interest formula

time in years

continuously compounding interest formula

A=Pe^rt

"P" in the continuously compounding interest formula

starting value

"r" in the continuously compounding interest formula

interest rate

"t" in the continuously compounding interest formula

time in years

Half-life Formula

y=a(.5)^(t/h)

"a" in the half-life formula

starting value

"t" in the half-life formula

time of measurement

"h" in the half-life formula

half-life

exponential form of logarithms

y=b^x

logarithm form

logb^y=x

log. loop

b raised to the x = y

common logarithm

when there is no given base, assume it is 10

power property of logs

LOGbM^k to kLOGbM

quotient property of logs

LOGb(M/N) to LOGbM-LOGbN

product property of logs

LOGb(MN) to LOGbM+LOGbN

exponential equations with variables equation

a=b^cx

solving exponential equations w/ variables

-move the exponent in front of the log by the power rule

-take the log of both sides to get x by istelf

-take the log of both sides to get x by istelf

change of base formula

LOGbM to LOG10M / LOG10b

increases toward infinity

In a standard exponential function y = 2^x, when x increases toward infinity, y ___.

increases toward zero

In a standard exponential function y = 2^x, when x decreases towards negative infinity, y ___.

e

number with a value of 2.718

exponential function

y = a^x

c units to the right

f(x) = a^(x-c) shifts f(x) = a^x ___

c units to the left

f(x) = a^(x+c) shifts f(x) = a^x ___

c units up

f(x) = a^(x) + c shifts f(x) = a^x ___

c units down

f(x) = a^(x) - c shifts f(x) = a^x ___

y = c

the horizontal asymptote of f(x) = a^(x) +/- c

flip over x-axis

f(x) = -a^x will make f(x) = a^x ___

flip over y-axis

f(x) = a^(-x) shifts f(x) = a^x ___

bases are the same

you can solve exponential equation only if the ____

logarithms

exponents

exponential function

___ is the inverse of a logarithm

exponential function

y = 2^x

logarithmic functions

x = 2^y

b

base value

e

exponent

a

answer

positive

the b and a values must be ___

positive or negative

the e value can be ___

1

the b value cannot equal ___

x

log v b B^x

x

b^(log v b (x))

log v b m + log v b n

log v b mn

log v b m - log v b n

log v b (m/n)

(x) log v b m

log v b m^x

Logarithm - Definition

The inverse of taking the exponent of something

log₃27 =3

3³ =27

log₈1/64 = -2

8⁻² = 1/64

log₃27 =3

3³ =27

log₈1/64 = -2

8⁻² = 1/64

Logarithm Property 1

log₂8 + log₂32 = log₂256

Logarithm Property 2

log₃1/9 - log₃81 = log₃1/729

Logarithm Property 3

3 ∗ log₂8 = log₂ of 8³

Logarithm Property 4

log₁₇357 = log₁₀357/log₁₀17

Logarithm Property 5

= log₂√32/√8

= log₂(32/√8)¹/²

=1/2 ∗ log₂(32/√8)

= 1/2(log₂32 -1/2 ∗ log₂8)

= log₂32 - 1/4 ∗ log₂8

= 5/2 - 3/4 = 7/4

= log₂(32/√8)¹/²

=1/2 ∗ log₂(32/√8)

= 1/2(log₂32 -1/2 ∗ log₂8)

= log₂32 - 1/4 ∗ log₂8

= 5/2 - 3/4 = 7/4

LogxAB=logxA+logxB

EX:

Log₄192=3.7925

EX:

Log₄192=3.7925

Product Property

LogxA/B=LogxA-LogxB

Quotient Property

LogbMp=PlogGbM

EX:log₂25=4.6438

EX:log₂25=4.6438

Power Property

if logxA=logxB, than a=b

EX:log₆x+log₆(x-9)=2

log₆x(x-9)=2

x(x-9)=6²

x²-9x-36=0

(x-12)(x+3)=0

x-12=0 or x+3=0

x=12 x=-3

Check which anwser works in the problem

in the calculator.

EX:log₆x+log₆(x-9)=2

log₆x(x-9)=2

x(x-9)=6²

x²-9x-36=0

(x-12)(x+3)=0

x-12=0 or x+3=0

x=12 x=-3

Check which anwser works in the problem

in the calculator.

Equality Property

Log10x=Logx

SUPER EASY DO IN CALCULATOR

SUPER EASY DO IN CALCULATOR

Common Logarithms

LogxY=logY/LogX

EX.

4^x=19

log4^X=log19

xlog4=log19

x=log19/log4

x≈2.1240

EX.

4^x=19

log4^X=log19

xlog4=log19

x=log19/log4

x≈2.1240

Change of base formula

A=PErt

A=Amount after T years

P=Principal (original) amount intrested

R=Annual interest rate

EX.

A=3000e^(0.04)(10)

A=3000e^0.4

A≈$4475.47

DON'T FORGET TO PUT THE DECIMAL PLACES INTO THE HUNDREDTH PLACE FOR MONEY!!!!

A=Amount after T years

P=Principal (original) amount intrested

R=Annual interest rate

EX.

A=3000e^(0.04)(10)

A=3000e^0.4

A≈$4475.47

DON'T FORGET TO PUT THE DECIMAL PLACES INTO THE HUNDREDTH PLACE FOR MONEY!!!!

Compound Interest Formula

4e^-2x -5=3

4e^-2x=8

e^-2x=2

ln e^-2x=ln 2

-2x=ln 2

x=ln2/-2

x≈-0.3466

4e^-2x=8

e^-2x=2

ln e^-2x=ln 2

-2x=ln 2

x=ln2/-2

x≈-0.3466

...

TRY THIS:

3e^4x -12=15

3e^4x -12=15

Base E

For any positive number u, except 1

y = ᵤᶢ → logᵤy = ᶢ

When y > 0, When u > 0, u ≠ 1

logᵤy = Defined

Defined

logᵤu = 1

Defined

logᵤ1 = 0

If u and x are positive, a ≠ 1

logᵤxʳ = rlogᵤx

If u,x,y are positive, a ≠ 1

logᵤxy = logᵤx + logᵤy

If u,x,y are positive, a ≠ 1

logᵤ(x/y) = logᵤx - logᵤy

For any number r

r = rlogᵤu = logᵤuʳ

If i,r,u are positive, i ≠ 1, u ≠ 1

logᵢr = logᵤr/logᵤi

If i,r,u are positive, i ≠ 1, u ≠ 1

logᵢr = 1/logᵣi

exponential function

A function with a variable in the exponent. Have horizontal asymptotes.

logarithmic function

The inverse of an exponential function. Have vertical asymptotes.

property of equality

you use this when solving an equation: you must do the same thing to both sides of an equation

expression

a group of symbols that make a mathematical statement

equation

A mathematical sentence that contains an equals sign.

argument of a log

must be greater than zero

common logarithms

logarithms with base 10

natural logs

logarithms with base e

The natural exponential function

f(x) = e^x

The Euler number

approximately 2.718

example of polynomial function

f(x)=3x^4 - 2x^3 + 3x -5

rational function

a function that is expressed as the quotient of 2 polynomial functions.

T-chart for graphing log functions

1,0, base, 1

T-chart for graphing exponential functions

0, 1, 1, base

change to exponential form: log 100 = 2

10^2 = 100

change to log form: e^0 = 1

ln 1 = 0

log 1 =

0

ln e =

1

ln 1

1

anything raised to the zero power is what?

1

when you multiply like bases you do what to the exponents?

add

when you divide like bases you do what to the exponents?

subtract

when you add like terms you do what to the exponent?

nothing

exponential growth

the base is an integer and greater than 1 with no reflection.

exponential decay

has a base between 0 and 1 (a fraction) with no reflection.

exponential growth

may have a base between 0 and 1 if there IS a reflection

a square root

is a power of 1/2

the pivot point

is the top numbers in the T-chart after transformations

to find the y-intercept for exponential functions

put 0 in for x and solve

compounding continuously

A=Pe^rt

compounding a specific # of times per year

A=P(1+r/n)^nt

doubling

A is 2 and P is 1

half life

A is .5 and P is 1

decay

A = pe^(-kt)

calculator input

put parenthesis around the entire denominator and around the entire power.

Logarithm of X

power to which a base number must be raised to equal X

log

base-10

ln

base-e

log (mn)

log m + log n

log (m/n)

log m - log n

log (m^n)

(n) log m

log(b)1

0

log(b)b

1

log(b)b^x

x

b^log(b)x

x

log1

0

log10

1

log10^x

x

10^logx

x

ln1

0

lne

1

lne^x

x

e^lnx

x

The Product Rule: log(b)(MN)

log(b)N+log(b)M

The Quotient Rule: log(b)(M/N)

log(b)N-log(b)M

The Power Rule: log(b)M^p

plog(b)M

Change of Base (common logs): log(b)M

logM/logb

Change of Base (natural logs): log(b)M

lnM/lnb

The exponential function with base b

f(x)=b^x

The natural exponential function

f(x)=e^x

Formula for interest compounded n times per year

A=P(1+r/n)^(nt)

A= balance

P= prinicpal

t= time in years

n= number of times compounded

r= interest rate (in decimal form)

A= balance

P= prinicpal

t= time in years

n= number of times compounded

r= interest rate (in decimal form)

Formula for interest compounded continuously

A=Pe^(rt)

A= balance

P= prinicpal

t= time in years

r= interest rate (in decimal form)

A= balance

P= prinicpal

t= time in years

r= interest rate (in decimal form)

Express in logarithmic form: y=b^x

log(b)x=y

Express in exponential form: log(b)x=y

y=b^x

e

An irrational number called the natural base

e is about 2.7183

e is about 2.7183

The common log logx means:

log(10)x

The natural log lnx means:

log(e)x

Describe the transformation f(x)=b^(x)+c

Shifts the graph c units upward

Describe the transformation f(x)=b^(x)-c

Shifts the graph c units downward

Describe the transformation f(x)=b^(x+c)

Shifts the graph c units left

Describe the transformation f(x)=b^(x-c)

Shifts the graph c units right

Describe the transformation f(x)=-b^x

Reflects the graph about the x-axis

Describe the transformation f(x)=b^(-x)

Reflects the graph about the y-axis

Describe the transformation f(x)=cb^x

Vertical Stretch c>1

Vertial Shrink 0<c<1

Multiply the y-coordinate of each point by "c"

Vertial Shrink 0<c<1

Multiply the y-coordinate of each point by "c"

log(b)N+log(b)M

log(b)(MN)

log(b)N-log(b)M

log(b)(M/N)

plog(b)M

log(b)M^p

what does n^5 mean?

That n is multiplied by itself 4 times (n x n x n x n x n); note this is not 5 because n^1 is multiplied x 1, n^2 is multiplied by itself once (n x n), etc.

a^m x a^n =

a^ (m+n) // to multiply numbers with the same base, keep the base and ADD the indices -p. 386

a^m / a^n =

a^(m-n) // page 386

(a^m)^n =

a^(mxn) // page 386

(ab)^n =

a^n x b^n // the power of a product is the product of the powers; p. 387

(a/b)^n

a^n / b^n -387

a^0 =

1 as long as a is not zero in which case it's zero

a^(-1) =

1/a // remember that an exponent changes its sign when you move from numerator to denominator and back again -387

a^(1/n) =

nth root of a // see p. 389 for the notation; note that the square root of a number is the same as that number raised to the power of 1/2.

square root of a can also be expressed as a to the power of ...

a^(1/2) -389

What is 8^(2/3) [8 to the power of two-thirds]?

8^(2/3) = (8^(1/3))^2 = 2^2 = 4 -390

exponential function definition

a function in which the variable occurs as part of the index or exponent; simplest have the form f(x) = a^x where a is positive and not equal to 1.

2^(-x) =

(1/2) ^ x [391]

When you put money in the bank, what do you call your balance? What do you call the money the bank pays you? What do you call the effect of this growth?

Your balance is your PRINCIPAL; what the bank pays you is INTEREST; how the money grows is COMPOUNDED since the interest is then itself added to the principal which then earns you more interest -395

What is the formula for the future value of an amount initially invested (present value) compounded at an interest rate per year of i for n years?

Future Value = Present Value x (1 + i) ^ n [396]

If you invest $5000 at 8% for 2 years, how much money will you have?

FV = PV x (1+i)^n

FV = 5000 x (1 + .08) ^ 2 = 5000 x (1.08)^2 = $5,832 -396

FV = 5000 x (1 + .08) ^ 2 = 5000 x (1.08)^2 = $5,832 -396

If I must save $10,000 for a boat in 4 years, and can get 8.5% per year return on my initial investment, how much should I invest today?

FV = PV x (1+i) ^n

10,000 = PV x (1.085)^4

Solve for PV => PV = $7,215.74

10,000 = PV x (1.085)^4

Solve for PV => PV = $7,215.74

What is the formula for depreciation?

Future Value = Present Value x (1 - i) ^ n [399] This is the mirror image of the future value of an amount compounded, it's just that a depreciation rate is just a NEGATIVE interest rate.

If a^x = a^k, what do we know about x and k?

x = k [page 400]

What is a logarithm?

A logarithm in base 10 of any positive number is the power you would have to raise 10 to get that number. So the log of 100 is 2 since 10^2 is 100. The log of 1,000 is 3 and the log of 1,000,000 is 6 (for this special case, you can just count the zeros). -404

What is the log of 1?

0

What is the log of 100?

2

What is the log of 1000?

3

log(b)1

0

log(b)b

1

log(b)b^x

x

b^log(b)x

x

log1

0

log10

1

log10^x

x

10^logx

x

ln1

0

lne

1

lne^x

x

e^lnx

x

The Product Rule: log(b)(MN)

log(b)N+log(b)M

The Quotient Rule: log(b)(M/N)

log(b)N-log(b)M

The Power Rule: log(b)M^p

plog(b)M

Change of Base (common logs): log(b)M

logM/logb

Change of Base (natural logs): log(b)M

lnM/lnb

The exponential function with base b

f(x)=b^x

The natural exponential function

f(x)=e^x

Formula for interest compounded n times per year

A=P(1+r/n)^(nt)

A= balance

P= prinicpal

t= time in years

n= number of times compounded

r= interest rate (in decimal form)

A= balance

P= prinicpal

t= time in years

n= number of times compounded

r= interest rate (in decimal form)

Formula for interest compounded continuously

A=Pe^(rt)

A= balance

P= prinicpal

t= time in years

r= interest rate (in decimal form)

A= balance

P= prinicpal

t= time in years

r= interest rate (in decimal form)

Express in logarithmic form: y=b^x

log(b)x=y

Express in exponential form: log(b)x=y

y=b^x

e

An irrational number called the natural base

e is about 2.7183

e is about 2.7183

The common log logx means:

log(10)x

The natural log lnx means:

log(e)x

Describe the transformation f(x)=b^(x)+c

Shifts the graph c units upward

Describe the transformation f(x)=b^(x)-c

Shifts the graph c units downward

Describe the transformation f(x)=b^(x+c)

Shifts the graph c units left

Describe the transformation f(x)=b^(x-c)

Shifts the graph c units right

Describe the transformation f(x)=-b^x

Reflects the graph about the x-axis

Describe the transformation f(x)=b^(-x)

Reflects the graph about the y-axis

Describe the transformation f(x)=cb^x

Vertical Stretch c>1

Vertial Shrink 0<c<1

Multiply the y-coordinate of each point by "c"

Vertial Shrink 0<c<1

Multiply the y-coordinate of each point by "c"

log(b)N+log(b)M

log(b)(MN)

log(b)N-log(b)M

log(b)(M/N)

plog(b)M

log(b)M^p

Logarithm

the exponent required to produce a given number, this of a positive number y to the base b is defined as follows: If y=b^x, then log b y=x