Exam 5: Exponential and Logarithmic Functions

The demand function for a limited edition comic book is given by $p = 3000 \left( 1 - \frac { 5 } { 5 + e ^ { - 0.015 x } } \right)$ Find the price $p$ for a demand of $x = 45$ units.

Sketch the graph of the function $g ( x ) = 4 ^ { x }$ .

Approximate the solution of $16 e ^ { 7 x } = 22$ to 3 decimal places. (You may use a graphing utility.)

Apply the Inverse Property of logarithmic or exponential functions to simplify the expression below. $\log _ { 8 } 64 ^ { 2 x + 5 }$

On the Richter scale, the magnitude $R$ of an earthquake of intensity $I$ per unit of area is given by $R = \log _ { 10 } \left( \frac { I } { I _ { 0 } } \right)$ where $I _ { 0 } = 1$ is the minimum intensity used for comparison. Find the magnitude $R$ (on the Richter scale) of an earthquake of intensity $I = 8,710,000$ Round to the nearest thousandth.

Determine whether $e = \frac { 271,801 } { 99,990 }$ . Justify your answer.

Find the exact value of $\log _ { 5 } \sqrt [ 3 ] { 25 }$ without using a calculator.

The population P of a bacteria culture is modeled by $P = 3300 e ^ { k t }$ , where t is the time in hours. If the population of the culture was 5800 after 40 hours, how long does it take for the population to double? Round to the nearest tenth of an hour.

The half-life of 134Ba is 28.7 hours. How much of a 11.2 g sample of 134Ba will be left after 24.00 hours? Round to the nearest hundredth of a gram.

Write the exponential equation $e ^ { 3 / 2 } = 4.4817 \ldots$ in logarithmic form.

Use the functions given by $f ( x ) = \frac { 1 } { 8 } x - 3$ and $g ( x ) = x ^ { 2 }$ to find the value $\left( g ^ { - 1 } \circ g ^ { - 1 } \right) ( - 4 )$

Use the properties of logarithms to simplify the logarithmic expression below. $\log _ { 5 } \sqrt { 175 }$

Evaluate the logarithm $\log _ { 2 } \frac { 1 } { 32 }$ without using a calculator.

Match the function $f ( x ) = 2 ^ { - x }$ with its graph.

An initial investment of $4000 grows at an annual interest rate of 4% compounded continuously. How long will it take to double the investment?

Approximate the logarithm below using the properties of logarithms, given $\log _ { b } 2 \approx 0.3562,$ $\log _ { b } 3 \approx 0.5646,$ and $\log _ { b } 5 \approx 0.8271.$ $\log _ { b } \frac { 3 } { 2 }$

Write the logarithmic equation $\ln 6 = 1.792 \ldots$ in exponential form.

Solve for $x.$ $\ln ( 6 x - 11 ) = 0$

The chemical acidity of a solution is measured in units of pH: $\mathrm { pH } = - \log \left[ \mathrm { H } ^ { + } \right]$ , where [H+] is the hydrogen ion concentration in the solution. What is [H+] if the $\mathrm { pH } = 5.8 ?$

A company's profit $P$ for producing $x$ units is given by $P ( x ) = 47 x - 5736$ . Find the inverse function $P ^ { - 1 } ( x )$ and explain what it represents. Describe the domains of $P ( x )$ and $P ^ { - 1 } ( x )$ .