# 4.8 Fitting exponential models to data  (Page 11/12)

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 x f(x) 1 409.4 2 260.7 3 170.4 4 110.6 5 74 6 44.7 7 32.4 8 19.5 9 12.7 10 8.1
 x f(x) 0.15 36.21 0.25 28.88 0.5 24.39 0.75 18.28 1 16.5 1.5 12.99 2 9.91 2.25 8.57 2.75 7.23 3 5.99 3.5 4.81

logarithmic; $\text{\hspace{0.17em}}y=16.68718-9.71860\mathrm{ln}\left(x\right)$

 x f(x) 0 9 2 22.6 4 44.2 5 62.1 7 96.9 8 113.4 10 133.4 11 137.6 15 148.4 17 149.3

## Practice test

The population of a pod of bottlenose dolphins is modeled by the function $\text{\hspace{0.17em}}A\left(t\right)=8{\left(1.17\right)}^{t},$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is given in years. To the nearest whole number, what will the pod population be after $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ years?

About $\text{\hspace{0.17em}}13\text{\hspace{0.17em}}$ dolphins.

Find an exponential equation that passes through the points and

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 $\text{\hspace{0.17em}}6.25%\text{\hspace{0.17em}}$ APR, compounding daily, in order to reach his goal in $\text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ years? $1,947$ An investment account was opened with an initial deposit of$9,600 and earns $\text{\hspace{0.17em}}7.4%\text{\hspace{0.17em}}$ interest, compounded continuously. How much will the account be worth after $\text{\hspace{0.17em}}15\text{\hspace{0.17em}}$ years?

Graph the function $\text{\hspace{0.17em}}f\left(x\right)=5{\left(0.5\right)}^{-x}\text{\hspace{0.17em}}$ and its reflection across the y -axis on the same axes, and give the y -intercept.

y -intercept:

The graph shows transformations of the graph of $\text{\hspace{0.17em}}f\left(x\right)={\left(\frac{1}{2}\right)}^{x}.\text{\hspace{0.17em}}$ What is the equation for the transformation?

Rewrite $\text{\hspace{0.17em}}{\mathrm{log}}_{8.5}\left(614.125\right)=a\text{\hspace{0.17em}}$ as an equivalent exponential equation.

${8.5}^{a}=614.125$

Rewrite $\text{\hspace{0.17em}}{e}^{\frac{1}{2}}=m\text{\hspace{0.17em}}$ as an equivalent logarithmic equation.

Solve for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ by converting the logarithmic equation $\text{\hspace{0.17em}}lo{g}_{\frac{1}{7}}\left(x\right)=2\text{\hspace{0.17em}}$ to exponential form.

$x={\left(\frac{1}{7}\right)}^{2}=\frac{1}{49}$

Evaluate $\text{\hspace{0.17em}}\mathrm{log}\left(\text{10,000,000}\right)\text{\hspace{0.17em}}$ without using a calculator.

Evaluate $\text{\hspace{0.17em}}\mathrm{ln}\left(0.716\right)\text{\hspace{0.17em}}$ using a calculator. Round to the nearest thousandth.

$\mathrm{ln}\left(0.716\right)\approx -0.334$

Graph the function $\text{\hspace{0.17em}}g\left(x\right)=\mathrm{log}\left(12-6x\right)+3.$

State the domain, vertical asymptote, and end behavior of the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{5}\left(39-13x\right)+7.$

Domain: $\text{\hspace{0.17em}}x<3;\text{\hspace{0.17em}}$ Vertical asymptote: $\text{\hspace{0.17em}}x=3;\text{\hspace{0.17em}}$ End behavior: $\text{\hspace{0.17em}}x\to {3}^{-},f\left(x\right)\to -\infty \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x\to -\infty ,f\left(x\right)\to \infty$

Rewrite $\text{\hspace{0.17em}}\mathrm{log}\left(17a\cdot 2b\right)\text{\hspace{0.17em}}$ as a sum.

Rewrite $\text{\hspace{0.17em}}{\mathrm{log}}_{t}\left(96\right)-{\mathrm{log}}_{t}\left(8\right)\text{\hspace{0.17em}}$ in compact form.

${\mathrm{log}}_{t}\left(12\right)$

Rewrite $\text{\hspace{0.17em}}{\mathrm{log}}_{8}\left({a}^{\frac{1}{b}}\right)\text{\hspace{0.17em}}$ as a product.

Use properties of logarithm to expand $\text{\hspace{0.17em}}\mathrm{ln}\left({y}^{3}{z}^{2}\cdot \sqrt[3]{x-4}\right).$

$3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ln}\left(y\right)+2\mathrm{ln}\left(z\right)+\frac{\mathrm{ln}\left(x-4\right)}{3}$

Condense the expression $\text{\hspace{0.17em}}4\mathrm{ln}\left(c\right)+\mathrm{ln}\left(d\right)+\frac{\mathrm{ln}\left(a\right)}{3}+\frac{\mathrm{ln}\left(b+3\right)}{3}\text{\hspace{0.17em}}$ to a single logarithm.

Rewrite $\text{\hspace{0.17em}}{16}^{3x-5}=1000\text{\hspace{0.17em}}$ as a logarithm. Then apply the change of base formula to solve for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ using the natural log. Round to the nearest thousandth.

$x=\frac{\frac{\mathrm{ln}\left(1000\right)}{\mathrm{ln}\left(16\right)}+5}{3}\approx 2.497$

Solve $\text{\hspace{0.17em}}{\left(\frac{1}{81}\right)}^{x}\cdot \frac{1}{243}={\left(\frac{1}{9}\right)}^{-3x-1}\text{\hspace{0.17em}}$ by rewriting each side with a common base.

Use logarithms to find the exact solution for $\text{\hspace{0.17em}}-9{e}^{10a-8}-5=-41$ . If there is no solution, write no solution .

$a=\frac{\mathrm{ln}\left(4\right)+8}{10}$

Find the exact solution for $\text{\hspace{0.17em}}10{e}^{4x+2}+5=56.\text{\hspace{0.17em}}$ If there is no solution, write no solution .

Find the exact solution for $\text{\hspace{0.17em}}-5{e}^{-4x-1}-4=64.\text{\hspace{0.17em}}$ If there is no solution, write no solution .

no solution

Find the exact solution for $\text{\hspace{0.17em}}{2}^{x-3}={6}^{2x-1}.\text{\hspace{0.17em}}$ If there is no solution, write no solution .

Find the exact solution for $\text{\hspace{0.17em}}{e}^{2x}-{e}^{x}-72=0.\text{\hspace{0.17em}}$ If there is no solution, write no solution .

$x=\mathrm{ln}\left(9\right)$

Use the definition of a logarithm to find the exact solution for $\text{\hspace{0.17em}}4\mathrm{log}\left(2n\right)-7=-11$

Use the one-to-one property of logarithms to find an exact solution for $\text{\hspace{0.17em}}\mathrm{log}\left(4{x}^{2}-10\right)+\mathrm{log}\left(3\right)=\mathrm{log}\left(51\right)\text{\hspace{0.17em}}$ If there is no solution, write no solution .

$x=±\frac{3\sqrt{3}}{2}$

The formula for measuring sound intensity in decibels $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ is defined by the equation $\text{\hspace{0.17em}}D=10\mathrm{log}\left(\frac{I}{{I}_{0}}\right),$ where $\text{\hspace{0.17em}}I\text{\hspace{0.17em}}$ is the intensity of the sound in watts per square meter and $\text{\hspace{0.17em}}{I}_{0}={10}^{-12}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}4.7\cdot {10}^{-1}\text{\hspace{0.17em}}$ watts per square meter?

A radiation safety officer is working with $\text{\hspace{0.17em}}112\text{\hspace{0.17em}}$ grams of a radioactive substance. After $\text{\hspace{0.17em}}17\text{\hspace{0.17em}}$ days, the sample has decayed to $\text{\hspace{0.17em}}80\text{\hspace{0.17em}}$ 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?

$f\left(t\right)=112{e}^{-.019792t};$ half-life: about $\text{\hspace{0.17em}}35\text{\hspace{0.17em}}$ days

Write the formula found in the previous exercise as an equivalent equation with base $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ Express the exponent to five significant digits.

A bottle of soda with a temperature of $\text{\hspace{0.17em}}\text{71°}\text{\hspace{0.17em}}$ Fahrenheit was taken off a shelf and placed in a refrigerator with an internal temperature of After ten minutes, the internal temperature of the soda was 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?

$T\left(t\right)=36{e}^{-0.025131t}+35;T\left(60\right)\approx {43}^{\text{o}}\text{F}$

The population of a wildlife habitat is modeled by the equation $\text{\hspace{0.17em}}P\left(t\right)=\frac{360}{1+6.2{e}^{-0.35t}},$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ 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.

 x f(x) 1 3 2 8.55 3 11.79 4 14.09 5 15.88 6 17.33 7 18.57 8 19.64 9 20.58 10 21.42

logarithmic

The population of a lake of fish is modeled by the logistic equation $\text{\hspace{0.17em}}P\left(t\right)=\frac{16,120}{1+25{e}^{-0.75t}},$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is time in years. To the nearest hundredth, how many years will it take the lake to reach $\text{\hspace{0.17em}}80%\text{\hspace{0.17em}}$ of its carrying capacity?

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.

 x f(x) 1 20 2 21.6 3 29.2 4 36.4 5 46.6 6 55.7 7 72.6 8 87.1 9 107.2 10 138.1

exponential; $\text{\hspace{0.17em}}y=15.10062{\left(1.24621\right)}^{x}$

 x f(x) 3 13.98 4 17.84 5 20.01 6 22.7 7 24.1 8 26.15 9 27.37 10 28.38 11 29.97 12 31.07 13 31.43
 x f(x) 0 2.2 0.5 2.9 1 3.9 1.5 4.8 2 6.4 3 9.3 4 12.3 5 15 6 16.2 7 17.3 8 17.9

logistic; $\text{\hspace{0.17em}}y=\frac{18.41659}{1+7.54644{e}^{-0.68375x}}$

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim