Exam 4: Inverse, Exponential, and Logarithmic Functions

Solve the problem. -The number of reports of a certain virus has increased exponentially since year 0. The number of cases can be approximated using the function r(t) = 478 e0.002t, where t is the number of years Since year 0. Estimate the number of cases in year 40.

Decide whether or not the functions are inverses of each other.Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = x ^ { 3 } - 6 , \quad g ( x ) = \sqrt [ 3 ] { x + 6 }$

Graph the exponential function using transformations where appropriate. - $f(x)=-2^{x-1}$

Write the word or phrase that best completes each statement or answers the question. -Many states have passed laws against smoking in public places since January. The total number of states, $N$ , that have passed a no smoking law, $t$ months after January is given by the function $\mathrm{N}(\mathrm{t})=\frac{50}{1+18 \mathrm{e}^{-0.7 \mathrm{t}}}$ . How many states had passed a law in January? Round to the nearest thousandth.

Decide whether the given functions are inverses. - =\{(1,8),(8,-3),(-3,-9)\} g=\{(8,1),(-3,8),(-9,1)\}

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $f ( x ) = 6 ^ { x }$ . Find $f ( - 3 )$ .

Write an equivalent expression in exponential form. - $\log _ { 10 } 100,000 = 5$

Given $\log _ { 10 } 2 = 0.3010 \text { and } \log _ { 10 } 3 = 0.4771 \text {, }$ find the logarithm without using a calculator. - $\log _ { 10 } 72$

Write the word or phrase that best completes each statement or answers the question. -Given $f ( x ) = 5 ^ { x }$ , evaluate the following. (a) $f \left( \log _ { 5 } 3 \right)$ (b) $f \left[ \log _ { 5 } ( 4 \ln 5 ) \right]$ (c) $\mathrm { f } \left[ \log _ { 5 } ( \ln 5 ) \right]$

Solve the equation and express the solution in exact form. - $\ln \mathrm { e } ^ { \mathrm { x } } - \ln \mathrm { e } ^ { 4 } = \ln \mathrm { e } ^ { 5 }$

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

If the function is one-to-one, find its inverse. If not, write ʺnot one-to-one.ʺ - $\{ ( 7,2 ) , ( 8,2 ) , ( 9,2 ) , ( 10,5 ) \}$

Write the word or phrase that best completes each statement or answers the question. -Use the formula $\mathrm { D } = 10.0 \log \left( \mathrm { S } / \mathrm { S } _ { 0 } \right)$ , where the loudness of a sound in decibels is determined by $\mathrm { S }$ , the number of watt $/ \mathrm { m } ^ { 2 }$ produced by the soundwave, and $\mathrm { S } _ { 0 } = 1.00 \times 10 ^ { - 12 } \mathrm { watt } / \mathrm { m } ^ { 2 }$ . What is the intensity in watt/ $/ \mathrm { m } ^ { 2 }$ of a noise measured at 37 decibels? (Round to the nearest tenth.)

Write the word or phrase that best completes each statement or answers the question. -Assume the cost of a gallon of milk is $\$ 3.00$ . With continuous compounding, find the time it would take the cost to be 5 times as much (to the nearest tenth of a year), at an annual inflation rate of $6 \%$ .

Evaluate the logarithm. - $\log _ { 13 } 13$

Solve the equation. - $\log _ { x } 625 = 4$

Find the domain and range of the inverse of the given function. - $f ( x ) = \sqrt { x - 1 }$

Write the word or phrase that best completes each statement or answers the question. - $\text { Explain the error in the following: } \log _ { 6 } 8 - \log _ { 6 } N = \log _ { 6 } ( 8 - N ) \text {. }$

Provide an appropriate response. -The graph of an exponential function with base a is given. Is a > 1 or is 0 < a < 1? Explain how you arrived at your answer. Give the domain and range of f. $f ( x ) = a ^ { x }$

Write the word or phrase that best completes each statement or answers the question. -Given $\mathrm { f } ( \mathrm { x } ) = \ln x$ , evaluate the following. (a) $f \left( e ^ { 5 } \right)$ (b) $f \left( e ^ { \ln 3 } \right)$ (c) $f \left( e ^ { 3 } \ln 3 \right)$