Exam 8: Exponential and Logarithmic Functions and Applications

Verify that f and g are inverse functions by showing that $\text { (a) } ( f \circ g ) ( x ) = x \text { and } ( b ) ( g \circ f ) ( x ) = x \text {. }$ $f ( x ) = 2 x - 1 \text { and } g ( x ) = \frac { x + 1 } { 2 }$

Solve the exponential equation by taking the logarithm of both sides. $2 ^ { 4 - t } = e$

What is the definition of the natural log function?

An initial amount of $5500 is invested in an account with interest compounded continuously. The interest rate is 10%. Find the value of the account after 14 years.

Solve the exponential equation by using the property that bx = by implies x = y, for b > 0 and b≠ 1. $4 ^ { - x } = 256$

When $f ( x ) = x + 6$ and $g ( x ) = 3 x ^ { 2 } + 3 x$ , find $( f + g ) ( x )$ .

The amount of radioactive carbon-14 in an archaeological specimen is given by $C = C _ { 0 } e ^ { - 0.000121 t }$ where $C _ { 0 }$ is the original amount, and t is the number of years after the specimen died. How many years would it take for the carbon-14 remaining to be 80% of the original amount? Round to the nearest year.

The level of a sound in decibels is calculated using the formula $D = 10 \cdot \log \left( I \times 10 ^ { 12 } \right)$ where I is the intensity of the sound waves in watts per square meter. The sound intensity inside a typical car at 60 mph is 0.00001 watts per square meter. How many decibels is that?

Write the equation in exponential form. $y = \log _ { b } x$

Use a calculator to approximate the logarithm. Round to 4 decimal places. $\log 80$

In 2000, the population of Sheboygan, WI was about 113,000 and was growing at a rate of 0.85% per year. Use an exponential function to predict the population in 2021 to the nearest whole Number.

Evaluate without the use of a calculator. $\log 0.001$

Expand into a sum and/or difference of logarithms. Assume all variables represent positive real numbers. $\log _ { 8 } a ^ { 17 } b ^ { 8 }$

If $f ( x ) = 12 x + 7$ and $g ( x ) = x ^ { 2 } - 5 x$ , find $( f \cdot g ) ( x )$

The population of bacteria culture was 2000 at noon, and was increasing at a rate of 10% per hour. The number can be found using the function $P ( t ) = 2,000 ( 1.1 ) ^ { t }$ where t is the number of hours past noon. Predict the population 9 hours later, at 9 PM to the nearest Whole number.

If money is invested at an annual return rate of r%, the number of years it takes to triple in value is given by $t = \frac { 100 \ln 3 } { r }$ ( r expressed as a percent ) . How long would it take the value of a mutual fund to triple if it averages 13% growth per year? Round to the nearest tenth of a year.

Solve the exponential equation by using the property that bx = by implies x = y, for b > 0 and b≠ 1. $4 ^ { 1 + x } = \left( \frac { 1 } { 2 } \right) ^ { 9 - x }$

If $f ( x ) = 10 x + 7$ and $g ( x ) = x ^ { 2 } - 3 x$ , find $\left( \frac { f } { g } \right) ( x )$

Solve the exponential equation by taking the logarithm of both sides. $e ^ { 6 y } = 40$

Evaluate without the use of a calculator. $\log \left( 1 \times 10 ^ { 5 } \right)$