Exam 5: Inverse, Exponential, and Logarithmic Functions

Graph the function. Give the domain and range. - f(x)=

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers with $a \neq 1 \text { and } b \neq 1$ - $\log _ { 3 } t - \log _ { 3 } s$

Choose the one alternative that best completes the statement or answers the question. -In the formula N $\mathrm { N } = \mathrm { Ie } ^{\mathrm { kt }}$ , N is the number of items in terms of an initial population I at a given time t and k is a growth constant equal to the percent of growth per unit time. How long will it take for The population of a certain country to double if its annual growth rate is 6.3%? Round to the nearest Year.

Write an equivalent expression in exponential form. - $\log ( x + 4 ) 13 = 1$

Evaluate the logarithm. - $\log _ { 1 / 3 } 3$

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

Find the value. Give an approximation to four decimal places. - $\log 587 - \log 266$

Write the word or phrase that best completes each statement or answers the question. -Given $f ( x ) = 8 ^ { x }$ , evaluate the following. (a) $f \left( \log _ { 8 } 5 \right)$ (b) $\mathrm { f } \left[ \log _ { 8 } ( 8 \ln 8 ) \right]$ (c) $f \left[ \log _ { 8 } ( \ln 8 ) \right]$

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = 5 x - 5 , \quad g ( x ) = \frac { 1 } { 5 } x + 1$

Solve the following graphically. If necessary, round answers to the nearest thousandth. -The exact solution to an exponential equation such as $2 ^ { x } = 3$ can be expressed in three forms: $\log _ { 2 } 3 , \frac { \log 3 } { \log 2 }$ , or $\frac { \ln 3 } { \ln 2 }$ . Give the exact solution to the following exponential equation in three forms. $3 ^ { x } = 11$

Solve the problem. -The loudness of a sound can be quantified in units called decibels, where the number of decibels $\mathrm { d }$ is given by the formula $\mathrm { d } = 10 \log \frac { \mathrm { I } } { \mathrm { I } _ { 0 } }$ . What is the decibel rating of a sound having an intensity $\mathrm { I } = 1000 \mathrm { I } _ { 0 }$ ?

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { \sqrt { 6 } } 193.7$

Decide whether the given functions are inverses. - () 1 1 2 2 3 3 4 4 5 5 x g(x) 1 1 -2 2 3 3 4 4 5 5

Match the function with its graph. - $f ( x ) = \log _ { 3 } ( x - 3 )$

Write the word or phrase that best completes each statement or answers the question. -Why can't 1 be the base of a logarithmic function?

Solve the problem. -Which of the following is the same as $\log ( 12 x ) - \log ( 3 x )$ for $x > 0$ ?

Solve the equation and express the solution in exact form. - $\log ( x - 3 ) = 1 - \log x$

Write the word or phrase that best completes each statement or answers the question. -The graph of $\mathrm { y } = 10 ^ { \mathrm { x } }$ is shown with the coordinates of a point displayed at the bottom of the screen. Write the logarithmic equation associated with the display.

Use properties of logarithms to evaluate the expression. - $100 ^ { \log _ { 10 } 9 }$

Solve the problem. -The population of country A in millions is modeled by $f ( x ) = 17.5 e ^ { 0.0019 x }$ . During the same time period, the population of country B in millions is modeled by $\mathrm { g } ( \mathrm { x } ) = 13.5 \mathrm { e } ^ { 0.0107 \mathrm { x } }$ . In both formulas $\mathrm { x }$ is the number of years. Assuming these trends continue, estimate what the population will be when the populations are equal. Round to the nearest million.