Exam 10: Inverse, Exponential, and Logarithmic Functions

Solve the equation. Use natural logarithms. When appropriate, give solutions to three decimal places unless otherwise indicated. - $\mathrm{e}^{-0.17 \mathrm{t}}=0.09$

A city is growing at the rate of $0.5 \%$ annually. If there were $3,889,000$ residents in the city in 1994 , find how many (to the nearest ten-thousand) were living in that city in 2000. Use $\mathrm{y}=3,889,000(2.7)^{0.005 t}$

Use the horizontal line test to determine if the function is one-to-one. -

A classmate missed the lecture concerning the power rule of logarithms. When your classmate calls you on the phone for an explanation of why it works, what should you say to explain it?

Find the indicated value. -Let $f(x)=2^{x} \cdot f^{-1}(8)$

Find the amount of money in an account after 11 years if $\$ 1100$ is deposited at $8 \%$ annual interest compounded quarterly. Assume no money is withdrawn.

Rewrite the given expression as a single logarithm. Assume that all variables are defined in such a way that variable expressions are positive and bases are positive numbers not equal to 1 . - $5 \log _{q} q-\log _{q} r$

Explain how the graph of the function $g(x)=\log _{5} x$ can be obtained from the graph of the functionf $(x)=5^{x}$ .

Rewrite the given expression as a single logarithm. Assume that all variables are defined in such a way that variable expressions are positive and bases are positive numbers not equal to 1 . - $4 \log _{c} q-\frac{2}{5} \log _{c} r+\frac{1}{6} \log _{c} f-6 \log _{c} p$

$\log _{6} 60-\log _{6} 10=1$

The total number $N$ of rabbits on a farmer's property, assuming unlimited resources and space, can be approximated by the function $\mathrm{N}(\mathrm{x})=50 \mathrm{e}^{0.452 \mathrm{x}}$ , where $\mathrm{x}=0$ corresponds to the initial number of rabbits, and $x=1$ corresponds to the number of rabbits after one year, and so on. The function is graphed on a graphing calculator-generated screen. Interpret the meanings of $x$ and $y$ in the display at the bottom of the screen.

Express the given logarithm as a sum and/or difference of logarithms. Simplify, if possible. Assume that all variables represent positive real numbers. - $\log _{19} \sqrt{\frac{m n}{3}}$

Express the given logarithm as a sum and/or difference of logarithms. Simplify, if possible. Assume that all variables represent positive real numbers. - $\log _{4}(144 \cdot 227)$

Use properties of logarithms to write each expression as a single logarithm. Assume that variables represent positive real numbers, with base $\neq 1$ . - $6 \log _{c} m-\frac{2}{5} \log _{c} n+\frac{1}{4} \log _{c} j-5 \log _{c} k$

Determine whether or not the function is one-to-one. - $f(x)=x^{3}-1$

Use the horizontal line test to determine if the function is one-to-one. -

The number of bacteria growing in an incubation culture increases with time according to $B(x)=8800(4)^{x}$ , where $x$ is time in days. After how many days will the number of bacteria in the culture be 563,200 ?(Hint: Let $B(x)=563,200$ .)

Graph the function. - $f(x)=5 x$

Determine whether or not the function is one-to-one. -

Use properties of logarithms to write each expression as a single logarithm. Assume that variables represent positive real numbers, with base $\neq 1$ . - $3 \log _{b} q-\log _{b} r$