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. - $\log ( 5 + x ) - \log ( x - 5 ) = \log 3$

Solve the equation by expressing each side as a power of the same base and then equating exponents. - $\mathrm { e } ^ { \mathrm { x } + 9 } = \frac { 1 } { \mathrm { e } ^ { 3 } }$

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. - $2 \log x = \log 196$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 3 } \left( \frac { 7 \cdot 11 } { 13 } \right)$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $5 ^ { 3 x } = 2.8$

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. - $\log _ { 3 } \left( x ^ { 2 } - 2 x \right) = 1$

Evaluate the expression without using a calculator. -The long jump record, in feet, at a particular school can be modeled by $f ( x ) = 18.4 + 2.5 \ln ( x + 1 )$ where $x$ is the number of years since records began to be kept at the school. What is the record for the long jump 13 years after record started being kept? Round your answer to the nearest tenth.

Evaluate the expression without using a calculator. - $\log _ { 64 } 4$

Approximate the number using a calculator. Round your answer to three decimal places. - $6 ^ { 3.6 }$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $e ^ { x + 5 } = 2$

Solve the problem. -The size of the coyote population at a national park increases at the rate of $4.7 \%$ per year. If the size of the current population is 196 , find how many coyotes there should be in 6 years. Use $\mathrm { y } = \mathrm { y } _ { \mathrm { o } } \mathrm { e } ^ { 0.047 \mathrm { t } }$ and round to the nearest whole number.

Solve the equation by expressing each side as a power of the same base and then equating exponents. - $25 ^ { x + 8 } = 125 ^ { x - 3 }$

Solve the problem. -The logistic growth function $\mathrm { f } ( \mathrm { t } ) = \frac { 64,000 } { 1 + 1279 \mathrm { e } ^ { - 1.7 t } }$ models the number of people who have become ill with a particular infection $t$ weeks after its initial outbreak in a particular community. How many people were ill after 9 weeks?

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { b } \left( y z ^ { 8 } \right)$

Solve. -The population of a certain country is growing at a rate of $1.4 \%$ per year. How long will it take for this country's population to double? Use the formula $t = \frac { \ln 2 } { k }$ , which gives the time, $t$ , for a population with growth rate $k$ , to double. (Round to the nearest whole year.)

Graph the function - $\text { Use the graph of } f ( x ) = e ^ { x } \text { to obtain the graph of } g ( x ) = e ^ { x / 2 } - 1$

Graph the function -Use the graph of $f ( x ) = 4 ^ { x }$ to obtain the graph of $g ( x ) = 4 \cdot 4 ^ { x }$ .

Write the equation in its equivalent logarithmic form. - $3 - 2 = \frac { 1 } { 9 }$

Solve the problem. -Find out how long it takes a $\$ 3000$ investment to double if it is invested at $7 \%$ compounded monthly. Round to the nearest tenth of a year. Use the formula $\mathrm { A } = \mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { n } } \right) ^ { \mathrm { nt } }$ .

Graph the function -Use the graph of $f ( x ) = 5 ^ { x }$ to obtain the graph of $g ( x ) = 5 ^ { x } + 1$