Exam 3: Exponential, Logistic, and Logarithmic Functions

Assuming all variables are positive, use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. - $\log _ { 3 } \left( \frac { x ^ { 5 } y ^ { 2 } } { 8 } \right)$

Decide whether the function is an exponential growth or exponential decay function and find the constant percentage rate of growth or decay. - $f ( x ) = 176 \cdot 2 ^ { x }$

Find the exact solution to the equation. - $\log _ { 2 } ( x - 4 ) = - 1$

State whether the function is an exponential growth function or exponential decay function, and describe its end behavior using limits. - $f ( x ) = 96 x$

Solve the problem. -A certain noise produces $3.89 \times 10 ^ { - 4 }$ Watts $/ \mathrm { m } ^ { 2 }$ of power. What is the decibel level of this noise? The level of sound intensity in decibels $( \mathrm { dB } )$ is $\beta = 10 \log \left( \frac { \mathrm { I } } { \mathrm { I } } \right)$ , where $\beta$ (beta) is the number of decibels, 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 decibel.)

Find the domain of the function. - $f ( x ) = \ln \left( 10 x - x ^ { 2 } \right)$

Describe how to transform the graph of the basic function g(x) into the graph of the given function f(x). - $f ( x ) = \ln ( - x ) - 4 ; g ( x ) = \ln x$

Find the exponential function that satisfies the given conditions. -Initial value $= 53$ , decreasing at a rate of $0.48 \%$ per week

Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. - $f ( x ) = 4 \cdot e ^ { - x }$

Use the product, quotient, and power rules of logarithms to rewrite the expression as a single logarithm. Assume that all variables represent positive real numbers. - $5 \log _ { 4 } ( 5 x - 5 ) + 6 \log _ { 4 } ( 3 x - 3 )$

Determine the doubling time of the investment. -5.86% APR compounded continuously

Solve the equation. - $3 ^ { - x } = \frac { 1 } { 9 }$

Solve the problem. -Find the periodic payment of a loan with present value $161,000 and an annual interest rate 5% for a term of 24 years, with payments made and interest charged 12 times per year.

Solve the problem. -Wind speed varies in the first twenty meters above the ground. For a particular day, let f(x) = 1.8 ln x + 7.2 compute the wind speed x meters above the ground. What is the wind speed 11 meters above the ground? Round results to the nearest hundredth.

Solve the problem. -Use the formula $\beta = 10 \log \left( \mathrm { I } _ { 0 } \right)$ , where the loudness of a sound in decibels is determined by I, the number of Watts/ $\mathrm { m } ^ { 2 }$ produced by the sound wave, and $\mathrm { I } _ { 0 } = 10 ^ { - 12 } \mathrm {~W} / \mathrm { m } ^ { 2 }$ . A certain noise measures 101 decibels. If the intensity is multiplied by 100 , how many decibels will the new noise measure? (Round to the nearest unit.)

Use a calculator to find an approximate solution to the equation. - $4 ( 4 x - 3 ) = 16$

Find the exact solution to the equation. - $8 - \log 5 ( x + 7 ) = 7$

Solve the problem. -Find the present value of a loan with an annual interest rate of 6.7% and periodic payments of $1266.21 for a term of 30 years, with payments made and interest charged 12 times per year.

Use the product, quotient, and power rules of logarithms to rewrite the expression as a single logarithm. Assume that all variables represent positive real numbers. - $7 \ln ( x y ) - 3 \ln ( y z )$

Graph the function. Describe its position relative to the graph of the indicated basic function. - $f ( x ) = 1 - e ^ { - 0.73 x } \text {; relative to } f ( x ) = e ^ { x }$