Exam 5: Logarithmic Functions

The half-life of a substance is 34 hours. If there are initially 125 grams of the substance, write an exponential formula to determine the amount $Q$ remaining after $t$ hours.

The following table gives $z$ as a linear function of $x$ . If $z=\ln u$ and $u=\ln y$ , find a formula for $y$ in terms of $x$ .

Find the domain of the function $f(x)=\log \left(x^{2}-16\right)-\log (-3 x)$ .

The following figure shows the graphs of (A). $y=e^{-x}$ (B). $y=10^{-x}$ (C). $y=\ln (-x)$ (D). $y=\log (-x)$ . Which one is the graph of A?

The following table gives $v$ as a linear function of $w$ . If $w=\ln x$ and $v=\ln y$ , find $y(x)$ at $x=15$ . Give your answer in exponential form.

The notation $\log _{b} x$ means that the $\log$ base is unspecified. Given that $\log _{b} x=3$ , $\log _{b} y=3.2$ , and $\log _{b} z=4.7$ , use properties of $\log$ to find $\log _{b}\left(\frac{x y^{3}}{z}\right)$ .

An earthquake measured 5.1 on the Richter scale. How many times more powerful were the seismic waves than standard seismic waves?

Use linear regression on the values $\mathrm{x}$ and $\ln y$ to fit a function of the form $\ln y=b+m x$ . Use a calculator program to find the regression line for this data. Convert to an exponential function $y=a e^{k x}$ and give the value of $a$ . Round to 4 decimal places.

Suppose the population of an endangered species is cut in half every 60 years. Assume exponential decay. A) What is the annual growth rate? B) What is the continuous growth rate? Give your answers correct to 4 decimal places.

Rewriting $e^{6 a}=b$ using logs gives

If the equation $Q=8 \cdot 3^{1.7 t}$ is converted to the form $Q=a e^{k t}$ , then $a=$ ----------- and $k=$ ------------- Give k to 4 decimal places.

Solve $58 e^{0.7 t}=7+t$ for $t$ to 2 decimal places if possible. If not possible, enter "not possible".

The vertical intercept of the graph of $y=e^{x-4}$ is at $y=\ldots$ . If necessary, round to 2 decimal places.

The following table gives $v$ as a linear function of $w$ . If $w=\ln x$ and $v=\ln y$ , find a formula for $y$ in terms of $x$ .

A certain country is experiencing deflation and prices are falling by $2.9 \%$ each month. If a certain item currently costs $\$ 31$ , how much will the item cost in a year?

The following table gives $w$ as a linear function of $u$ . If $w=\ln v$ , find $v(u)$ at $u=10$ . Give your answer in exponential form.

The doubling time for a bank account that is growing by $2.6 \%$ per year (compounded continually) is---------- years. Round 1 decimal place.

Rewriting the statement $\ln \left(\frac{1}{8}\right)=-2.079$ using exponents gives

Solve for $t$ : $600 e^{0.7 t}=500 e^{0.1 t}$ . Round your answer to 3 decimal places.

A town with an initial population of 14,000 people doubles every 40 years. Write an exponential equation to determine the town's population $P$ after $t$ years.