Exam 3: Exponential, Logistic, and Logarithmic Functions

Solve the problem. -A thermometer is removed from a cup of coffee and placed in water whose temperature $\left( \mathrm { T } _ { \mathrm { m } } \right)$ is $10 ^ { \circ } \mathrm { C }$ . The following data were collected over the next 30 seconds. Use an exponential regression equation for the $\mathrm { T } - \mathrm { T } \mathrm { m }$ data to estimate the thermometer reading when it was removed from the hot coffee.

Simplify the expression. - $\log _ { 4 } 4 ^ { 3 }$

Solve the problem. -The number of students infected with the flu on a college campus after $t$ days is modeled by the function $P ( t ) = \frac { 160 } { 1 + 39 e ^ { - 0.3 t } }$ . What was the initial number of infected students?

Find the following using a calculator. Round to four decimal places. - $\log 5$

Compute the exact value of the function for the given x-value without using a calculator. - $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$ for $x = 2$

Solve the problem. -How long must $4100 be in a bank at 8% compounded annually to become $10,324.50? (Round to the nearest year.)

Determine a formula for the exponential function. -The graph of an exponential function is given. Which of the following is the correct equation of the function?

Solve the problem. -The number of students infected with the flu on a college campus after $t$ days is modeled by the function $P ( t ) = \frac { 240 } { 1 + 39 e ^ { - 0.3 t } }$ . When will the number of infected students be 120 ?

Solve the problem. -What is the intensity in Watts/ $\mathrm { m } ^ { 2 }$ of a noise measured at 66 decibels? The level of sound intensity in decibels $( \mathrm { dB } )$ is $\beta = 10 \log \left( \mathrm { I } / \mathrm { I } _ { 0 } \right)$ , where $\beta$ (beta) is the number of decibels, $\mathrm { I }$ is the sound intensity in $\mathrm { W } / \mathrm { m } ^ { 2 }$ , and $\mathrm { I } _ { 0 } =$ $10 ^ { - 12 } \mathrm {~W} / \mathrm { m } ^ { 2 }$ . (Round to the nearest hundredth.)

Decide whether the function is an exponential growth or exponential decay function and find the constant percentage rate of growth or decay. - $f ( x ) = 1.5 \cdot 1.044 ^ { x }$

Compute the exact value of the function for the given x-value without using a calculator. - $f ( x ) = 5 ^ { x } \text { for } x = - 1$

Use a calculator to find an approximate solution to the equation. - $150 ( 1.52 ) ^ { x / 2 } = 300$

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

Describe how to transform the graph of the basic function g(x) into the graph of the given function f(x). - $f ( x ) = \log ( x ) + 2 ; \quad g ( x ) = \log x$

Match the function f with its graph. - $f ( x ) = \ln ( x )$

Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. - $f ( x ) = \ln x + 2$

Find the logistic function that satisfies the given conditions. -

Solve the problem. -The growth in the population of a certain rodent at a dump site fits the exponential function $\mathrm { A } ( \mathrm { t } ) = 862 \mathrm { e } 0.032 \mathrm { t }$ , where $t$ is the number of years since 1963. Estimate the population in the year 2000 .

Evaluate the logarithm. - $\log 0.0001$

Write the expression using only the indicated logarithms. -log5 10 using common logarithms