Exam 5: Inverse, Exponential, and Logarithmic Functions

The decibel level D of a sound is related to its intensity I by $\mathrm { D } = 10 \log \left( \frac { \mathrm { I } } { \mathrm { I } _ { 0 } } \right)$ . If $\mathrm { I } _ { 0 }$ is $10 ^ { - 12 }$ , then what is the intensity of a noise measured at 93 decibels? Express your answer in scientific notation, rounding to three significant digits, if necessary.

How long will it take for prices in the economy to double at a 7% annual inflation rate?

Write in logarithmic form. - $5 ^ { 2 } = 25$

Round your answer to the nearest tenth, when appropriate. Use the formula $\mathrm { pH } = - \log \left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$ , as needed. -Find $\left[ \mathrm { H } _ { 3 } \mathrm { O } ^ { + } \right]$ if the $\mathrm { pH } = 6$ .

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $\{ ( - 2,4 ) , ( 2 , - 4 ) , ( 8 , - 2 ) , ( - 8,2 ) \}$

Determine whether or not the function is one-to-one. - $f ( x ) = \sqrt { x + 2 }$

Assume the cost of a car is $21,000. With continuous compounding in effect, find the number of years it would take to double the cost of the car at an annual inflation rate of 6.2%. Round the answer to the nearest hundredth.

A sample of 400 grams of radioactive substance decays according to the function $\mathrm { A } ( \mathrm { t } ) = 400 \mathrm { e } ^ { - .035 \mathrm { t } }$ , where t is the time in years. How much of the substance will be left in the sample after 10 years? Round your answer to the Nearest whole gram.

Use your graphing calculator to find how long it will take for $9000 invested at 3.325% per year compounded daily to triple in value. Find the answer to the nearest year. Use 365 days for a year.

Find the function value. If the result is irrational, round your answer to the nearest thousandth. - $f ( x ) = 4 ^ { x }$

Given $\log _ { 10 } 2 \approx 0.3010 \text { and } \log _ { 10 } 3 \approx 0.4771$ find the logarithm without using a calculator. - $\log _ { 10 } \frac { 27 } { 4 }$

Write an equivalent expression in exponential form. - $\log _{( x + 1 )} 12 = 1$

Determine whether the statement is true or false. -If a function f has an inverse function and the domain of f is all real numbers, then the domain of $\mathrm { f } ^ { - 1 }$ must also be all real numbers.

Find the value. Give an approximation to four decimal places. -ln 43,100,000

The growth in the mouse population at a certain county dump can be modeled by the exponential function A(t)= 395e0.013t, where t is the number of months since the population was first recorded. Estimate the Population after 28 months.

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. $3 ^ { x } = 15$

Solve the equation and express the solution in exact form. - $\log _ { 9 } ( x - 5 ) + \log _ { 9 } ( x - 5 ) = 1$

Write an equivalent expression in exponential form. - $\log _ { 5 } 625 = 4$

Write an equivalent expression in exponential form. -x = log10 0.0000001

The amount of particulate matter left in solution during a filtering process is given by the equation $P = 700 ( 2 ) ^ { - 0.8 n }$ , where $\mathrm { n }$ is the number of filtering steps. Find the amounts left for $\mathrm { n } = 0$ and $\mathrm { n } = 5$ . (Round to the nearest whole number.)