6.8 Fitting Exponential Models to Data

Find the linear regression line using the cricket-chirp data in the example earlier in this section, and find the temperature if there are 30 chirps in 15 seconds.

the table shows a recent graduate’s credit card balance each month after graduation.

ⓐ Use exponential regression to fit a model to these data.
ⓑ If spending continues at this rate, what will the graduate’s credit card debt be one year after graduating?

Sales of a video game released in the year 2000 took off at first, but then steadily slowed as time moved on. the table shows the number of games sold, in thousands, from the years 2000–2010.

1. ⓐ Let $x$ represent time in years starting with $x = 1$ for the year 2000. Let $y$ represent the number of games sold in thousands. Use logarithmic regression to fit a model to these data.
2. ⓑ If games continue to sell a…

the table shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012.

1. ⓐ Let $x$ represent time in years starting with $x = 0$ for the year 1997. Let $y$ represent the number of seals in thousands. Use logistic regression to fit a model to these data.
2. ⓑ Use the model to predict the seal population for the year 2020.
3. ⓒ To the nearest whole number, what is the limiting va…

What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

What is regression analysis? Describe the process of performing regression analysis on a graphing utility.

What does the y-intercept on the graph of a logistic equation correspond to for a population modeled by that equation?

$y = 5.598 - 1.912 \ln \left(x\right)$

$y = 4.607 + 2.733 \ln \left(x\right)$

To the nearest whole number, what is the initial value of a population modeled by the logistic equation $P \left(t\right) = \frac{175}{1 + 6.995 e^{- 0.68 t}} ?$ What is the carrying capacity?

A logarithmic model is given by the equation $h \left(p\right) = 67.682 - 5.792 \ln \left(p\right) .$ To the nearest hundredth, for what value of $p$ does $h \left(p\right) = 62 ?$

What is the y-intercept on the graph of the logistic model given in the previous exercise?

Use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P \left(x\right) = \frac{68}{1 + 16 e^{- 0.28 x}} .$

What was the initial population of koi?

Use this scenario: The population $P$ of a koi pond over $x$ months is modeled by the function $P \left(x\right) = \frac{68}{1 + 16 e^{- 0.28 x}} .$

How many months will it take before there are $20$ koi in the pond?

Use this scenario: The population $P$ of an endangered species habitat for wolves is modeled by the function $P \left(x\right) = \frac{558}{1 + 54.8 e^{- 0.462 x}} ,$ where $x$ is given in years.

How many wolves will the habitat have after $3$ years?

Use this scenario: The population $P$ of an endangered species habitat for wolves is modeled by the function $P \left(x\right) = \frac{558}{1 + 54.8 e^{- 0.462 x}} ,$ where $x$ is given in years.

Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity.

Use the regression feature to find an exponential function that best fits the data in the table.

Write the exponential function as an exponential equation with base $e .$

Use the intersect feature to find the value of $x$ for which $f \left(x\right) = 250.$

Use the LOGarithm option of the REGression feature to find a logarithmic function of the form $y = a + b \ln \left(x\right)$ that best fits the data in the table.

Use the logarithmic function to find the value of the function when $x = 10.$

Use the intersect feature to find the value of $x$ for which $f \left(x\right) = 8.$

Use the LOGISTIC regression option to find a logistic growth model of the form $y = \frac{c}{1 + a e^{- b x}}$ that best fits the data in the table.

To the nearest whole number, what is the predicted carrying capacity of the model?

Use the intersect feature to find the value of $x$ for which the model reaches half its carrying capacity.

Use a graphing utility to find an exponential regression formula $f \left(x\right)$ and a logarithmic regression formula $g \left(x\right)$ for the points $\left(1.5 , 1.5\right)$ and $\left(8.5 , \text{8}.\text{5}\right) .$ Round all numbers to 6 decimal places. Graph the points and both formulas along with the line $y = x$ on the same axis. Make a conjecture about the relationship of the regression formulas.

Find the inverse function $f^{- 1} \left(x\right)$ for the logistic function $f \left(x\right) = \frac{c}{1 + a e^{- b x}} .$ Show all steps.

Determine whether the function $y = 156 \left(0.825\right)^{t}$ represents exponential growth, exponential decay, or neither. Explain

Find an exponential equation that passes through the points $(\text{2},\textrm{ }\text{2} .\text{25})$ and $\left(5 , 60.75\right) .$

A retirement account is opened with an initial deposit of $8,500 and earns $8.12 \%$ interest compounded monthly. What will the account be worth in $20$ years?

Does the equation $y = 2.294 e^{- 0.654 t}$ represent continuous growth, continuous decay, or neither? Explain.

The graph of $f \left(x\right) = 6.5^{x}$ is reflected about the y-axis and stretched vertically by a factor of $7.$ What is the equation of the new function, $g \left(x\right) ?$ State its y-intercept, domain, and range.

Rewrite $\log_{17} \left(4913\right) = x$ as an equivalent exponential equation.

Rewrite $a^{- \frac{2}{5}} = b$ as an equivalent logarithmic equation.

Solve for x if $\log_{64} \left(x\right) = \frac{1}{3}$ by converting the logarithmic equation $\log_{64} \left(x\right) = \frac{1}{3}$ to exponential form.

Evaluate $\log \left(0.000001\right)$ without using a calculator.

Evaluate $\ln \left(e^{- 0.8648}\right)$ without using a calculator.

State the domain, vertical asymptote, and end behavior of the function $g \left(x\right) = \ln \left(4 x + 20\right) - 17.$

Rewrite $\log_{8} \left(x\right) + \log_{8} \left(5\right) + \log_{8} \left(y\right) + \log_{8} \left(13\right)$ in compact form.

Rewrite $\ln \left(z\right) - \ln \left(x\right) - \ln \left(y\right)$ in compact form.

Rewrite $- \log_{y} \left(\frac{1}{12}\right)$ as a single logarithm.

Use properties of logarithms to expand $\ln \left(2 b \sqrt{\frac{b + 1}{b - 1}}\right) .$

Condense the expression $3 \log_{7} v + 6 \log_{7} w - \frac{\log_{7} u}{3}$ to a single logarithm.

Rewrite $5^{12 x - 17} = 125$ as a logarithm. Then apply the change of base formula to solve for $x$ using the common log. Round to the nearest thousandth.

Solve $\frac{125}{\left(\frac{1}{625}\right)^{- x - 3}} = 5^{3}$ by rewriting each side with a common base.

Use logarithms to find the exact solution for $3 e^{6 n - 2} + 1 = - 60.$ If there is no solution, write no solution.

Find the exact solution for $2 e^{5 x - 2} - 9 = - 56.$ If there is no solution, write no solution.

Find the exact solution for $e^{2 x} - e^{x} - 110 = 0.$ If there is no solution, write no solution.

Use the definition of a logarithm to find the exact solution for $9 + 6 \ln \left(a + 3\right) = 33.$

Use the one-to-one property of logarithms to find an exact solution for $\ln \left(5\right) + \ln \left(5 x^{2} - 5\right) = \ln \left(56\right) .$ If there is no solution, write no solution.

The population of a city is modeled by the equation $P \left(t\right) = 256 , 114 e^{0.25 t}$ where $t$ is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million?

Find the inverse function $f^{- 1}$ for the logarithmic function $f \left(x\right) = 0.25 \cdot \log_{2} \left(x^{3} + 1\right) .$

Use this scenario: A doctor prescribes $300$ milligrams of a therapeutic drug that decays by about $17 \%$ each hour.

Write an exponential model representing the amount of the drug remaining in the patient’s system after $t$ hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after $24$ hours. Round to the nearest hundredth of a gram.

Use this scenario: A soup with an internal temperature of $\text{350}°$ Fahrenheit was taken off the stove to cool in a $\text{71}°\text{F}$ room. After fifteen minutes, the internal temperature of the soup was $\text{175}°\text{F} .$

How many minutes will it take the soup to cool to $\text{85}°\text{F}?$

Use this scenario: The equation $N \left(t\right) = \frac{1200}{1 + 199 e^{- 0.625 t}}$ models the number of people in a school who have heard a rumor after $t$ days.

To the nearest tenth, how many days will it be before the rumor spreads to half the carrying capacity?

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

Find a formula for an exponential equation that goes through the points $\left(- 2 , 100\right)$ and $\left(0 , 4\right) .$ Then express the formula as an equivalent equation with base e.

The population of a culture of bacteria is modeled by the logistic equation $P \left(t\right) = \frac{14 , 250}{1 + 29 e^{- 0.62 t}} ,$ where $t$ is in days. To the nearest tenth, how many days will it take the culture to reach $75 \%$ of its carrying capacity?

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

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?

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

Solve for $x$ by converting the logarithmic equation $l o g_{\frac{1}{7}} \left(x\right) = 2$ to exponential form.

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

State the domain, vertical asymptote, and end behavior of the function $f \left(x\right) = \log_{5} \left(39 - 13 x\right) + 7.$

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

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

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

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

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

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

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

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?

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 $\text{35}°\text{ F} .$ After ten minutes, the internal temperature of the soda was $\text{63}°\text{ F} .$ 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?