Exam 4: Exponential and Logarithmic Functions

Use the One-to-One Property of Logarithms to Solve Logarithmic Equations 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. - $\log 3 x = \log 4 + \log ( x - 3 )$

Solve the problem. -Find out how long it takes a $\$ 2700$ investment to earn $\$ 300$ interest if it is invested at $7 \%$ compounded quarterly. Round to the nearest tenth of a year. Use the formula $\mathrm { A } = \mathrm { P } \left( 1 + \frac { \mathrm { r } } { \mathrm { n } } \right) ^ { \mathrm { n } t }$ .

Solve the problem. -The logistic growth function $\mathrm { f } ( \mathrm { t } ) = \frac { 57,000 } { 1 + 813.3 \mathrm { e } ^ { - 1.6 \mathrm { t } } }$ 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 became ill with this infection when the epidemic began?

Graph the function. -Use the graph of $f ( x ) = 2 ^ { x }$ to obtain the graph of $g ( x ) = 2 \cdot 2 ^ { x }$ .

Use Compound Interest Formulas Use the compound interest formulas A $A = P \left( 1 + \frac { r } { n } \right) \text { and } A = P e ^ { r t } \text { to solve }$ -Find the accumulated value of an investment of $1230 at 6% compounded annually for 5 years.

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $2 ^ { 5 x } = 3.7$

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. - $\log x + \log \left( x ^ { 2 } - 4 \right) - \log 3 - \log ( x - 2 )$

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. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. - $e ^ { 2 x } + e ^ { x } - 6 = 0$

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

Write the equation in its equivalent logarithmic form. - $c ^ { 4 } = 20,736$

Find the domain of the logarithmic function. - $f ( x ) = \log _ { 9 } ( x + 8 )$

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

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 7 } \left( \frac { \sqrt [ 5 ] { m } \sqrt [ 6 ] { n } } { k ^ { 2 } } \right)$

Solve the problem. -The logistic growth function $f ( t ) = \frac { 440 } { 1 + 5.3 e ^ { - 0.24 t } }$ describes the population of a species of butterflies t months after they are introduced to a non-threatening habitat. What is the limiting size of the butterfly population that the habitat will sustain?

Solve the problem. -Use the formula $\mathrm { R } = \log \left( \frac { \mathrm { a } } { \mathrm { T } } \right) + \mathrm { B }$ to find the intensity $\mathrm { R }$ on the Richter scale, given that amplitude a is 207 micrometers, time $\mathrm { T }$ between waves is $3.5$ seconds, and $\mathrm { B }$ is 2.7. Round answer to one decimal place.

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 3 } \left( \frac { 9 } { \sqrt { x - 1 } } \right)$

Use the Definition of a Logarithm to Solve Logarithmic Equations 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. - $\log _ { 2 } ( x + 1 ) - \log _ { 2 } ( x - 3 ) = 4$

Graph the function. -Use the graph of $f ( x ) = 3 ^ { x }$ to obtain the graph of $g ( x ) = 3 ^ { - x }$ .

Use Natural Logarithms Evaluate or simplify the expression without using a calculator. - $\mathrm { e } ^ { \ln 112 }$