Exam 4: Exponential and Logarithmic Functions

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log _ { 8 } x + \log _ { 8 } ( x - 63 ) = 2$

Solve the equation by expressing each side as a power of the same base and then equating exponents. - $4 ( 7 - 3 x ) = \frac { 1 } { 16 }$

Graph the function. -Use the graph of log2 x to obtain the graph of f(x) = -2 log2 x.

Solve the problem. -The population in a particular country is growing at the rate of $1.3 \%$ per year. If $2,546,000$ people lived there in 1999 , how many will there be in the year 2004 ? Use $\mathrm { y } = \mathrm { y } _ { \mathrm { o } } \mathrm { e } ^ { 0.013 \mathrm { t } }$ and round to the nearest ten-thousand.

The graph of a logarithmic function is given. Select the function for the graph from the options. -

Approximate the number using a calculator. Round your answer to three decimal places. -The function $f ( x ) = 500 ( 0.5 ) ^ { x / 100 }$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $x$ is the number of years since the material was put into the vault. Find the amount of radioactive material in the vault after 140 years. Round to the nearest whole number.

Graph the function by making a table of coordinates. - $f(x)=5^{x}$

Write the equation in its equivalent logarithmic form. - $b ^ { 2 } = 64$

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places - $\log _ { \pi } 22$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 3 } \left( \frac { x ^ { 5 } } { y ^ { 6 } } \right)$

Solve the problem. -The function $\mathrm { A } = \mathrm { A } _ { 0 } \mathrm { e } ^ { - 0.0099 \mathrm { x } }$ models the amount in pounds of a particular radioactive material stored in a concrete vault, where $x$ is the number of years since the material was put into the vault. If 800 pounds of the material are initially put into the vault, how many pounds will be left after 190 years?

Solve the problem. -The logistic growth function $f ( t ) = \frac { 65,000 } { 1 + 3249.0 e ^ { - 1.4 t } }$ models the number of people who have become ill with a particular infection $t$ weeks after its initial outbreak in a particular community. What is the limiting size of the population that becomes ill?

Evaluate the expression without using a calculator. - $\log _ { 5 } \frac { 1 } { 125 }$

Evaluate the expression without using a calculator. - $\ln e ^ { 5 x }$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { a } \left( \frac { x ^ { 4 } \sqrt [ 3 ] { x + 5 } } { ( x - 2 ) ^ { 2 } } \right)$

Solve. -The population of a particular country was 23 million in 1980 ; in 1990 , it was 34 million. The exponential growth function $\mathrm { A } = 23 \mathrm { e } ^ { \mathrm { kt } }$ describes the population of this country t years after 1980 . Use the fact that 10 years after 1980 the population increased by 11 million to find $\mathrm { k }$ to three decimal places.

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $\frac { 1 } { 2 } \log _ { 9 } x + \log _ { 9 } y$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $e ^ { 2 x } + e ^ { x } - 6 = 0$

Evaluate or simplify the expression without using a calculator. - $10 \log \sqrt [ 4 ] { x }$

Graph the functions in the same rectangular coordinate system. - $f(x)=\left(\frac{1}{3}\right)^{x} \text { and } g(x)=\log 1 / 3 x$