Exam 4: Exponential and Logarithmic Functions

Evaluate the expression. $\ln \left( e ^ { n + 1 } \right)$

A wooden mast salvaged from a shipwreck contains $95.5 \%$ of the carbon-14 that is present in living trees. How long ago was the ship built? (The half-life of carbon-14 is $5730$ years.) 

Evaluate the expression. $\log _ {5 } 625 - \log _ { 5 } 125$

Graph the function $y = 2 ^ { | x | }$ , not by plotting points but by applying your knowledge of the general shape of graphs of the form $a ^ { x }$ . State the domain, range, and asymptote of the function.

Use the change of base formula and a calculator to evaluate $\log _ { 91 } 19$ correct to six decimal places.

A culture starts with $680$ bacteria. After just $30$ minutes the count is $3250$ . Find a formula for the number of bacteria after $t$ hours. Find the number of bacteria after $3.5$ hours. After how many minutes will the number of bacteria triple? 

Use a calculator to evaluate the expression $\log 0.5 \text {, }$ to four decimal places.

Use a graphing calculator to find all solutions of the equation $\log _ {10 } x = 2 x ^ { 2 } - 5$ correct to two decimal places.

Use the Laws of Logarithms to rewrite the expression $\ln \frac { e ^ { 3 x } \sqrt { x ^ { 2 } + 1 } } { x ^ { 3 } + y ^ { 3 } }$ in a form with no logarithm of a product, quotient, root, or powers.  

For the function $y = \frac { 2 ^ { 2 / x } } { x }$ , ( $x < 0$ ), use a graphing calculator to find, correct to two decimal places, (a) the intervals on which the function is increasing or decreasing, and (b) the range of the function.

How long will it take an investment of $\$ 300$ to triple, if the interest rate is $3.5 \%$ per year compounded continuously? 

A radioactive substance decays in such a way that the amount of mass remaining after t days is given by the function $m ( t ) = 13 e ^ { - 0.015 t }$ where $m ( t )$ is measured in kilograms. Round your answers to three decimal places. a) Find the mass at time $t = 0$ . b) How much of the mass remains after 50 days?

If $10,000 is invested in an account at 5% per year, find the amount of the investment at the end of 12 years for the following compounding methods. (a) Continuously (b) Daily

The count in a culture of bacteria was $867$ after $3$ hours and $906$ after $4$ hours. (a) What is the relative rate of growth of the bacterial population? Express your answer as a percentage. (b) What was the initial size of the culture? (c) After how many hours with the bacteria reach $1200$ ? 

Evaluate the expression. $\log _ { 7 } 1$

A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time t (in hours) required for the colony to grow to N bacteria is given by $t = 3 \frac { \log ( N / 50 ) } { \log 2 }$ Find the time required for the colony to grow to two million bacteria.

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $h ( x ) = e ^ { - x } , \quad h ( 1 ) , h ( \sqrt { 2 } ) , h ( - 3 ) , h ( 1 / 2 )$

Find the solution of the equation $3 ^ { x + 2 } = 5 ^ { 2 x }$ correct to four decimal places.

Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals. $g ( x ) = ( 2 / 3 ) ^ { 2 x } ; g ( 0.7 ) , g ( \sqrt { 2 } / 2 ) , g ( 1 / \pi ) , g ( 1 / 3 )$

Sketch the graph of the function $h ( x ) = ( 0.6 ) ^ { x }$ by making a table of values. Use a calculator if necessary.