Exam 4: Inverse, Exponential, and Logarithmic Functions

Write the word or phrase that best completes each statement or answers the question. -In a town whose population is 3500 , a disease creates an epidemic. The number $\mathrm { N }$ of people infected $t$ days after the disease has begun is given by the function $N ( t ) = \frac { 3500 } { 1 + 15.6 \cdot e ^ { - 0.7 t } }$ . Find the number infected after 8 days.

Decide whether the given functions are inverses. -

The graph of a function f is given. Use the graph to find the indicated value. - $\mathrm { f } ^ { - 1 } ( 3 )$

Solve the problem. -The population of a particular city is increasing at a rate proportional to its size. It follows the function $\mathrm { P } ( \mathrm { t } ) = 1 + \mathrm { ke } ^ { 0.05 \mathrm { t } }$ where $\mathrm { k }$ is a constant and $\mathrm { t }$ is the time in years. If the current population is 36,000 , in how many years is the population expected to be 90,000 ? Round to the nearest year.

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $\ln ( x + 2 ) = - \sqrt [ 3 ] { x + 5 }$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 2 } 0.92$

Solve the problem. -The growth in population of a city can be seen using the formula $\mathrm { p } ( \mathrm { t } ) = 10,696 \mathrm { e } ^ { 0.008 \mathrm { t } }$ where t is the number of years. According to this formula, how many years will it take the population to double Its year 0 value? Round to the nearest tenth of a year.

Decide whether or not the functions are inverses of each other.Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = \sqrt { x + 2 } , x \geq - 2 ; \quad g ( x ) = x ^ { 2 } + 2$

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers. - $\frac { 3 } { 2 } \log _ { n } 9 y + \frac { 9 } { 5 } \log _ { n } \left( 81 y ^ { 2 } \right)$

Solve the problem. -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. $5 ^ { \mathrm { X } } = 12$

Provide an appropriate response. -The graph of an exponential function with base a is given. Sketch the graph of $g ( x ) = - a ^ { x }$ Give the domain and range of g. $f(x)=a^{x}$

Match the function with its graph. - $f ( x ) = \log _ { 4 } \left( \frac { 1 } { x } \right)$

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 5 } \left( \frac { 8 \sqrt { m } } { n } \right)$

Write the word or phrase that best completes each statement or answers the question. - $\text { Explain the error in the following: } \log _ { 4 } 3 \mathrm { y } = \log _ { 4 } 3 \cdot \log _ { 4 } \mathrm { y } \text {. }$

Write the word or phrase that best completes each statement or answers the question. -An economist predicts that the buying power $B ( x )$ of a dollar x years from now will decrease according to the formula $B ( x ) = 0.2 ^ { x }$ . How much will today's dollar be worth in 4 years? Round the answer to the nearest cent.

Solve the problem. -If $f ( x ) = \log 7 x$ , find $f \left( 7 ^ { 3 } \right)$ .

An everyday activity is described. Keeping in mind that an inverse operation ʺundoesʺ what an operation does, describe each inverse activity. -waking up

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 2 } 52.25$

Determine the function value. -Suppose f(x) = logax and f(4) = 2. Find f(16).

Write the word or phrase that best completes each statement or answers the question. -The height in meters of women of a certain tribe is approximated by $h = 0.52 + 2 \log ( \mathrm { t } / 3 )$ where $t$ is the woman's age in years and $1 \leq \mathrm { t } \leq 20$ . Estimate the height (to the nearest hundredth) of a woman of the tribe 4 years of age. (Round to the nearest hundredth.)