Given that $\log _a 2=0.301, \log _a 5=0.699$, and $\log _a 6=0.778$, find each of the following.

$\log _a \frac{1}{5}$

In your own words, explain the ambiguity of the title. Does the poem as a whole support just one meaning or all meanings?

A $0.33\text{-}\mu\text{F}$ capacitor is in parallel with a $0.15\text{-}\mu\text{F}$ and a $220,000\text{-pF}$ capacitor. What is the total capacitance, $C_\text{T}$? a. $0.7\space\mu\text{F}$. b. $0.007\space\mu\text{F}$. c. $0.07\space\mu\text{F}$. d. $7\text{ nF}$.

A dockworker applies a constant horizontal force of 80.0 N to a block of ice on a smooth horizontal floor. The frictional force is negligible. The block starts from rest and moves 11.0 m in 5.00 s. (a) What is the mass of the block of ice? (b) If the worker stops pushing at the end of 5.00 s, how far does the block move in the next 5.00 s?

Solve each equation.

$$\log _4 x-\log _4(x+3)=\log _4(x-2)$$

Find the distance between the two points.

(6,-1),(-9,8)

Determine whether the statement is true or false if

$A=\{6,10,12\},$

$B=\{5,9,11\}$,

$C=\{\ldots,-3,-2,-1,0,1,2,3, \ldots\},$

$$D=\{2,4,6, \ldots, 16\}$$

$E=\{x \mid x$ is a rational number $\},$

$$F=\{\},$$

$G=\{x \mid x$ is an irrational number $\}$,$and$H={x \mid x$is a real number$}$.$C \nsubseteq B$

State whether each equation is always true, sometimes true, or never true. Assume that $x$ is a positive real number.

$$\frac{\log 3}{\log x}=\log 3-\log x$$

Which of the following sets are equal?

$$\begin{aligned} & A=\left\{x \mid x^2-4 x+3=0\right\}, \quad C=\{x \mid x \in \mathbf{N}, x<3\}, \quad E=\{1,2\}, \quad G=\{3,1\}, \\ & B=\left\{x \mid x^2-3 x+2=0\right\}, \quad D=\{x \mid x \in \mathbf{N}, x \text { is odd, } x<5\}, \quad F=\{1,2,1\}, \quad H=\{1,1,3\} . \\ & \end{aligned}$$

Solve for n.

$$\log _b 8+3 \log _b n=3 \log _b(x-1)$$

Is the statement in the following problem true or false? Give an explanation for your answer.

A positive increasing exponential function always becomes larger than any increasing linear function in the long run.

Solve. Write the solution set in set-builder notation.

$9 x-2<7$ and $3 x-5>10$

A light spring with force constant 3.85 N/m is compressed by 8.00 cm as it is held between a 0.250-kg block on the left and a 0.500-kg block on the right, both resting on a horizontal surface. The spring exerts a force on each block, tending to push them apart. The blocks are simultaneously released from rest. Find the acceleration with which each block starts to move, given that the coefficient of kinetic friction between each block and the surface is 0.100.

In the following Exercise, a pair of power functions having the same coefficients but different exponents is given. In the long run, which function in the pair dominates?

$f(x)=10 \sqrt{x}$ and $g(x)=10 \sqrt[3]{x}$