# 6.8 Fitting exponential models to data  (Page 10/12)

 Page 10 / 12

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

## Logarithmic Functions

Rewrite $\text{\hspace{0.17em}}{\mathrm{log}}_{17}\left(4913\right)=x\text{\hspace{0.17em}}$ as an equivalent exponential equation.

${17}^{x}=4913$

Rewrite $\text{\hspace{0.17em}}\mathrm{ln}\left(s\right)=t\text{\hspace{0.17em}}$ as an equivalent exponential equation.

Rewrite $\text{\hspace{0.17em}}{a}^{-\text{\hspace{0.17em}}\frac{2}{5}}=b\text{\hspace{0.17em}}$ as an equivalent logarithmic equation.

${\mathrm{log}}_{a}b=-\frac{2}{5}$

Rewrite $\text{\hspace{0.17em}}{e}^{-3.5}=h\text{\hspace{0.17em}}$ as an equivalent logarithmic equation.

Solve for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{log}}_{64}\left(x\right)=\frac{1}{3}\text{\hspace{0.17em}}$ to exponential form.

$x={64}^{\frac{1}{3}}=4$

Evaluate $\text{\hspace{0.17em}}{\mathrm{log}}_{5}\left(\frac{1}{125}\right)\text{\hspace{0.17em}}$ without using a calculator.

Evaluate $\text{\hspace{0.17em}}\mathrm{log}\left(\text{0}\text{.000001}\right)\text{\hspace{0.17em}}$ without using a calculator.

$\mathrm{log}\left(\text{0}\text{.000001}\right)=-6$

Evaluate $\text{\hspace{0.17em}}\mathrm{log}\left(4.005\right)\text{\hspace{0.17em}}$ using a calculator. Round to the nearest thousandth.

Evaluate $\text{\hspace{0.17em}}\mathrm{ln}\left({e}^{-0.8648}\right)\text{\hspace{0.17em}}$ without using a calculator.

$\mathrm{ln}\left({e}^{-0.8648}\right)=-0.8648$

Evaluate $\text{\hspace{0.17em}}\mathrm{ln}\left(\sqrt[3]{18}\right)\text{\hspace{0.17em}}$ using a calculator. Round to the nearest thousandth.

## Graphs of Logarithmic Functions

Graph the function $\text{\hspace{0.17em}}g\left(x\right)=\mathrm{log}\left(7x+21\right)-4.$

Graph the function $\text{\hspace{0.17em}}h\left(x\right)=2\mathrm{ln}\left(9-3x\right)+1.$

State the domain, vertical asymptote, and end behavior of the function $\text{\hspace{0.17em}}g\left(x\right)=\mathrm{ln}\left(4x+20\right)-17.$

Domain: $\text{\hspace{0.17em}}x>-5;\text{\hspace{0.17em}}$ Vertical asymptote: $\text{\hspace{0.17em}}x=-5;\text{\hspace{0.17em}}$ End behavior: as $\text{\hspace{0.17em}}x\to -{5}^{+},f\left(x\right)\to -\infty \text{\hspace{0.17em}}$ and as $\text{\hspace{0.17em}}x\to \infty ,f\left(x\right)\to \infty .$

## Logarithmic Properties

Rewrite $\text{\hspace{0.17em}}\mathrm{ln}\left(7r\cdot 11st\right)\text{\hspace{0.17em}}$ in expanded form.

Rewrite $\text{\hspace{0.17em}}{\mathrm{log}}_{8}\left(x\right)+{\mathrm{log}}_{8}\left(5\right)+{\mathrm{log}}_{8}\left(y\right)+{\mathrm{log}}_{8}\left(13\right)\text{\hspace{0.17em}}$ in compact form.

${\text{log}}_{8}\left(65xy\right)$

Rewrite $\text{\hspace{0.17em}}{\mathrm{log}}_{m}\left(\frac{67}{83}\right)\text{\hspace{0.17em}}$ in expanded form.

Rewrite $\text{\hspace{0.17em}}\mathrm{ln}\left(z\right)-\mathrm{ln}\left(x\right)-\mathrm{ln}\left(y\right)\text{\hspace{0.17em}}$ in compact form.

$\mathrm{ln}\left(\frac{z}{xy}\right)$

Rewrite $\text{\hspace{0.17em}}\mathrm{ln}\left(\frac{1}{{x}^{5}}\right)\text{\hspace{0.17em}}$ as a product.

Rewrite $\text{\hspace{0.17em}}-{\mathrm{log}}_{y}\left(\frac{1}{12}\right)\text{\hspace{0.17em}}$ as a single logarithm.

${\text{log}}_{y}\left(12\right)$

Use properties of logarithms to expand $\text{\hspace{0.17em}}\mathrm{log}\left(\frac{{r}^{2}{s}^{11}}{{t}^{14}}\right).$

Use properties of logarithms to expand $\text{\hspace{0.17em}}\mathrm{ln}\left(2b\sqrt{\frac{b+1}{b-1}}\right).$

$\mathrm{ln}\left(2\right)+\mathrm{ln}\left(b\right)+\frac{\mathrm{ln}\left(b+1\right)-\mathrm{ln}\left(b-1\right)}{2}$

Condense the expression $\text{\hspace{0.17em}}5\mathrm{ln}\left(b\right)+\mathrm{ln}\left(c\right)+\frac{\mathrm{ln}\left(4-a\right)}{2}\text{\hspace{0.17em}}$ to a single logarithm.

Condense the expression $\text{\hspace{0.17em}}3{\mathrm{log}}_{7}v+6{\mathrm{log}}_{7}w-\frac{{\mathrm{log}}_{7}u}{3}\text{\hspace{0.17em}}$ to a single logarithm.

${\mathrm{log}}_{7}\left(\frac{{v}^{3}{w}^{6}}{\sqrt[3]{u}}\right)$

Rewrite $\text{\hspace{0.17em}}{\mathrm{log}}_{3}\left(12.75\right)\text{\hspace{0.17em}}$ to base $\text{\hspace{0.17em}}e.$

Rewrite $\text{\hspace{0.17em}}{5}^{12x-17}=125\text{\hspace{0.17em}}$ as a logarithm. Then apply the change of base formula to solve for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ using the common log. Round to the nearest thousandth.

$x=\frac{\frac{\mathrm{log}\left(125\right)}{\mathrm{log}\left(5\right)}+17}{12}=\frac{5}{3}$

## Exponential and Logarithmic Equations

Solve $\text{\hspace{0.17em}}{216}^{3x}\cdot {216}^{x}={36}^{3x+2}\text{\hspace{0.17em}}$ by rewriting each side with a common base.

Solve $\text{\hspace{0.17em}}\frac{125}{{\left(\frac{1}{625}\right)}^{-x-3}}={5}^{3}\text{\hspace{0.17em}}$ by rewriting each side with a common base.

$x=-3$

Use logarithms to find the exact solution for $\text{\hspace{0.17em}}7\cdot {17}^{-9x}-7=49.\text{\hspace{0.17em}}$ If there is no solution, write no solution .

Use logarithms to find the exact solution for $\text{\hspace{0.17em}}3{e}^{6n-2}+1=-60.\text{\hspace{0.17em}}$ If there is no solution, write no solution .

no solution

Find the exact solution for $\text{\hspace{0.17em}}5{e}^{3x}-4=6\text{\hspace{0.17em}}$ . If there is no solution, write no solution .

Find the exact solution for $\text{\hspace{0.17em}}2{e}^{5x-2}-9=-56.\text{\hspace{0.17em}}$ If there is no solution, write no solution .

no solution

Find the exact solution for $\text{\hspace{0.17em}}{5}^{2x-3}={7}^{x+1}.\text{\hspace{0.17em}}$ If there is no solution, write no solution .

Find the exact solution for $\text{\hspace{0.17em}}{e}^{2x}-{e}^{x}-110=0.\text{\hspace{0.17em}}$ If there is no solution, write no solution .

$x=\mathrm{ln}\left(11\right)$

Use the definition of a logarithm to solve. $\text{\hspace{0.17em}}-5{\mathrm{log}}_{7}\left(10n\right)=5.$

47. Use the definition of a logarithm to find the exact solution for $\text{\hspace{0.17em}}9+6\mathrm{ln}\left(a+3\right)=33.$

$a={e}^{4}-3$

Use the one-to-one property of logarithms to find an exact solution for $\text{\hspace{0.17em}}{\mathrm{log}}_{8}\left(7\right)+{\mathrm{log}}_{8}\left(-4x\right)={\mathrm{log}}_{8}\left(5\right).\text{\hspace{0.17em}}$ If there is no solution, write no solution .

Use the one-to-one property of logarithms to find an exact solution for $\text{\hspace{0.17em}}\mathrm{ln}\left(5\right)+\mathrm{ln}\left(5{x}^{2}-5\right)=\mathrm{ln}\left(56\right).\text{\hspace{0.17em}}$ If there is no solution, write no solution .

$x=±\frac{9}{5}$

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 large orchestra with a sound intensity of $\text{\hspace{0.17em}}6.3\cdot {10}^{-3}\text{\hspace{0.17em}}$ watts per square meter?

show that the set of all natural number form semi group under the composition of addition
what is the meaning
Dominic
explain and give four Example hyperbolic function
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Abdullahi
hi mam
Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak