Graph each function. $y=f(x)=3^{x+2}$

Graph $y=2^{x}$ and $y=\frac{2^{x}}{8}$ on the same screen. It appears that the graph of $y=\frac{2^{x}}{8}$ is the graph of $y=2^{x}$ shifted three units to the right. Prove algebraically that this is true.

Display the graph of the equation $(6) 5^{y}+x=2.$

We have seen that there are two kinds of bases, b, for exponential and logarithmic functions: those b in (0, 1) and those b in $(1, \infty).$ It might be supposed that there are more of the second kind but this is not the case. Consider the function $f:(0,1) \rightarrow (1, \infty)$ given by f(x) = 1/x. (a) Show that the domain of f can be taken to be (0, 1). (b) Show that with domain (0, 1) the range of f is $(1, \infty).$ (c) Show that $f$ has an inverse g and determine a formula for g(x). The exercise shows that the numbers in (0, 1) are in one-to-one correspondence with the numbers in $(1, \infty)$ so that every base of either kind corresponds to exactly one of the other kind. Who would have thought it? $``(1, \infty)-$so many numbers; (0, 1)-so little space."

Solve $6^{3-4 x}=15.$ Round your answer to two decimal places.

Solve $\ln x=6-2 x.$ Round your answer to two decimal places.

Determine the points of intersection of the graphs of $y=\ln x$ and y = x - 2 with coordinates rounded to two decimal places.

If $y=f(x)=\frac{\ln x}{x},$ determine the range of f. Round values to two decimal places.

Determine the x-values of points of intersection of the graphs of $y=(x-2)^{2}$ and $y=\ln x.$ Round your answers to two decimal places.

Use the graph of $y=\ln x$ to estimate $e^{2}$ to two decimal places.

An alternative to straight-line depreciation is declining-balance depreciation. This method assumes that an item loses value more steeply at the beginning of its life than later on. A fixed percentage of the value is subtracted each month. Suppose an item's initial cost is C and its useful life is N months. Then the value, V (in dollars), of the item at the end of n months is given by $V=C\left(1-\frac{1}{N}\right)^{n}$ so that each month brings a depreciation of $\frac{100}{N}$ percent. (This is called single declining-balance depreciation; if the annual depreciation were $\frac{200}{N}$ percent, then we would speak of double-declining-balance depreciation.) A notebook computer is purchased for $1500 and has a useful life of 36 months. It undergoes double-declining-balance depreciation. After how many months, to the nearest integer, does its value drop below$500?

Express each logarithmic form exponentially and each exponential form logarithmically. $\ln 3 \approx 1.09861$

Graph each function. $y=f(x)=3^{x+2}$

Let $\log 2=a, \log 3=b,$ and $\log 5=c.$ Express the indicated logarithm in terms of a, b, and c. $\log 100$