4.8 Fitting exponential models to data  (Page 12/12)

The population of a pod of bottlenose dolphins is modeled by the function $A(t)=8{(1.17)}^t,$ where $t$ is given in years. To the nearest whole number, what will the pod population be after $3$ years?

Find an exponential equation that passes through the points $\text{(0, 4)}$ and $\text{(2, 9)}\text{.}$

Drew wants to save $2,500 to go to the next World Cup. To the nearest dollar, how much will he need to invest in an account now with $6.25\%$ APR, compounding daily, in order to reach his goal in $4$ years?

An investment account was opened with an initial deposit of $9,600 and earns $7.4\%$ interest, compounded continuously. How much will the account be worth after $15$ years?

Graph the function $f(x)=5{\left(0.5\right)}^{-x}$ and its reflection across the y -axis on the same axes, and give the y -intercept.

The graph shows transformations of the graph of $f(x)={\left(\frac{1}{2}\right)}^x.$ What is the equation for the transformation?

Rewrite ${\log }_{8.5}\left(614.125\right)=a$ as an equivalent exponential equation.

Rewrite $e^{\frac{1}{2}}=m$ as an equivalent logarithmic equation.

Solve for $x$ by converting the logarithmic equation $lo{g}_{\frac{1}{7}}(x)=2$ to exponential form.

Evaluate $\log (\text{10,000,000})$ without using a calculator.

Evaluate $\ln \left(0.716\right)$ using a calculator. Round to the nearest thousandth.

Graph the function $g(x)=\log \left(12-6x\right)+3.$

State the domain, vertical asymptote, and end behavior of the function $f(x)={\log }_5\left(39-13x\right)+7.$

Rewrite $\log \left(17a\cdot 2b\right)$ as a sum.

Rewrite ${\log }_t\left(96\right)-{\log }_t\left(8\right)$ in compact form.

Rewrite ${\log }_8\left(a^{\frac{1}{b}}\right)$ as a product.

Use properties of logarithm to expand $\ln \left(y^3{z}^2\cdot \sqrt[3]{x-4}\right).$

Condense the expression $4\ln \left(c\right)+\ln \left(d\right)+\frac{\ln \left(a\right)}{3}+\frac{\ln \left(b+3\right)}{3}$ to a single logarithm.

Rewrite ${16}^{3x-5}=1000$ as a logarithm. Then apply the change of base formula to solve for $x$ using the natural log. Round to the nearest thousandth.

Solve ${\left(\frac{1}{81}\right)}^x\cdot \frac{1}{243}={\left(\frac{1}{9}\right)}^{-3x-1}$ by rewriting each side with a common base.

Use logarithms to find the exact solution for $-9e^{10a-8}-5=-41$ . If there is no solution, write no solution .

Find the exact solution for $10e^{4x+2}+5=56.$ If there is no solution, write no solution .

Find the exact solution for $-5e^{-4x-1}-4=64.$ If there is no solution, write no solution .

Find the exact solution for $2^{x-3}=6^{2x-1}.$ If there is no solution, write no solution .

Find the exact solution for $e^{2x}-e^x-72=0.$ If there is no solution, write no solution .

Use the definition of a logarithm to find the exact solution for $4\log \left(2n\right)-7=-11$

Use the one-to-one property of logarithms to find an exact solution for $\log \left(4x^2-10\right)+\log \left(3\right)=\log \left(51\right)$ If there is no solution, write no solution .

The formula for measuring sound intensity in decibels $D$ is defined by the equation $D=10\log \left(\frac{I}{I_0}\right),$ where $I$ is the intensity of the sound in watts per square meter and $I_0={10}^{-12}$ is the lowest level of sound that the average person can hear. How many decibels are emitted from a rock concert with a sound intensity of $4.7\cdot {10}^{-1}$ watts per square meter?

A radiation safety officer is working with $112$ grams of a radioactive substance. After $17$ days, the sample has decayed to $80$ grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest day, what is the half-life of this substance?

Write the formula found in the previous exercise as an equivalent equation with base $e.$ Express the exponent to five significant digits.

A bottle of soda with a temperature of $\text{71°}$ Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of $\text{35° F}\text{.}$ After ten minutes, the internal temperature of the soda was $\text{63° F}\text{.}$ Use Newton’s Law of Cooling to write a formula that models this situation. To the nearest degree, what will the temperature of the soda be after one hour?

The population of a wildlife habitat is modeled by the equation $P\left(t\right)=\frac{360}{1+6.2e^{-0.35t}},$ where $t$ is given in years. How many animals were originally transported to the habitat? How many years will it take before the habitat reaches half its capacity?

Enter the data from [link] into a graphing calculator and graph the resulting scatter plot. Determine whether the data from the table would likely represent a function that is linear, exponential, or logarithmic.

The population of a lake of fish is modeled by the logistic equation $P(t)=\frac{16,120}{1+25e^{-0.75t}},$ where $t$ is time in years. To the nearest hundredth, how many years will it take the lake to reach $80\%$ of its carrying capacity?