Exam 4: Exponential and Logarithmic Functions

Solve the problem. -The Richter scale converts seismographic readings into numbers for measuring the magnitude of an earthquake according to this function $M ( x ) = \log \left( \frac { x } { x _ { 0 } } \right) \text {, where } x _ { 0 } = 10 ^ { - 3 }$ What would be the readings x (to the nearest tenth) for magnitudes of 4.5 and 7.5?

Use a calculator to find the natural logarithm correct to four decimal places. - $\ln \sqrt { 17 }$

Find the effective rate of interest. -11.8% compounded continuously

Use a graphing calculator to solve the equation. Round your answer to two decimal places. - $e ^ { x } - \ln x = 4$

Solve the problem. -The half-life of plutonium-234 is 9 hours. If 60 milligrams is present now, how much will be present in 3 days? (Round your answer to three decimal places.)

Solve the problem. -A music store manager collected data regarding price and quantity demanded of cassette tapes every week for 10 weeks, and found that the exponential function of best fit to the data was p = 25 ·0.89q. Express the function of best fit in the form $\mathrm { p } = \mathrm { p } _ { 0 } \mathrm { e } ^ { \mathrm { kq } }$ , and use this expression to predict the quantity demanded if the price is $8.50.

Solve the problem. -The logistic growth functi $f ( t ) = \frac { 50,000 } { 1 + 2,499.0 e ^ { - 1.9 t } }$ 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?

For the given functions f and g, find the requested composite function. -f(x) = -2x + 3, g(x) = 6x + 4; Find (g °f)(x).

The loudness of a sound of intensity x, measured in watts per square meter, is defined as L( $L ( x ) = \log \left( \frac { x } { x _ { 0 } } \right) , \text { where } x _ { 0 } = 10 ^ { - 3 }$ -You have two friends, Jim and Amy. Jim always yells when he speaks, and Amy always whispers. The loudness of Jim's voice is 120 decibels, and the loudness of Amy's voice is 20 decibels. Determine how many times as intense Jim's voice is as compared at Amy's.

Change the exponential expression to an equivalent expression involving a logarithm. - $5 ^ { 2 } = 25$

Solve the equation. - $\mathrm { e } ^ { \mathrm { x } - 5 } = \left( \frac { 1 } { \mathrm { e } ^ { 5 } } \right) ^ { \mathrm { x } + 6 }$

Solve the problem. -The number of books in a small library increases according to the functio $B = 2,800 e ^ { 0.03 t }$ , where t is measured in years. How many books will the library have after 10 years?

Change the logarithmic expression to an equivalent expression involving an exponent. - $\ln \frac { 1 } { \mathrm { e } ^ { 5 } } = - 5$

For the given functions f and g, find the requested composite function value. - $f ( x ) = 4 x + 6 , \quad g ( x ) = 4 x ^ { 2 } + 3 ; \quad$ Find $( g \circ f ) ( 3 )$

Solve the equation. - $\left( e ^ { x } \right) ^ { x } \cdot e ^ { 54 } = e ^ { 15 x }$

Solve the problem. -The amount of a radioactive substance present, in grams, at time t in months is given by the formula $y = 8,000 ( 3 ) ^ { - 0.2 t }$ Find the number of grams present in 2 years. If necessary, round to three decimal places.

Solve the equation. - $\log _ { 4 } ( x + 2 ) = 2 + \log _ { 4 } ( x - 4 )$

Change the logarithmic expression to an equivalent expression involving an exponent. - $\log _ { b } 49 = \frac { 2 } { 3 }$

Write as the sum and/or difference of logarithms. Express powers as factors. - $\log _ { 8 } \frac { \sqrt [ 7 ] { x } \sqrt [ 3 ] { y } } { z ^ { 2 } }$

Solve the problem. -The surface area of a balloon is given by $\mathrm { S } ( \mathrm { r } ) = 4 \pi \mathrm { r } ^ { 2 }$ , where $\mathrm { r }$ is the radius of the balloon. If the radius is increasing with time $t$ , as the balloon is being blown up, according to the formula $r ( t ) = \frac { 2 } { 3 } t { } ^ { 3 } , t \geq 0$ , find the surface area $S$ as a function of the time $t .$