# 4.3 Logarithmic functions  (Page 6/9)

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${\mathrm{log}}_{15}\left(a\right)=b$

${15}^{b}=a$

${\mathrm{log}}_{y}\left(137\right)=x$

${\mathrm{log}}_{13}\left(142\right)=a$

${13}^{a}=142$

$\text{log}\left(v\right)=t$

$\text{ln}\left(w\right)=n$

${e}^{n}=w$

For the following exercises, rewrite each equation in logarithmic form.

${4}^{x}=y$

${c}^{d}=k$

${\text{log}}_{c}\left(k\right)=d$

${m}^{-7}=n$

${19}^{x}=y$

${\mathrm{log}}_{19}y=x$

${x}^{-\text{\hspace{0.17em}}\frac{10}{13}}=y$

${n}^{4}=103$

${\mathrm{log}}_{n}\left(103\right)=4$

${\left(\frac{7}{5}\right)}^{m}=n$

${y}^{x}=\frac{39}{100}$

${\mathrm{log}}_{y}\left(\frac{39}{100}\right)=x$

${10}^{a}=b$

${e}^{k}=h$

$\text{ln}\left(h\right)=k$

For the following exercises, solve for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ by converting the logarithmic equation to exponential form.

${\text{log}}_{3}\left(x\right)=2$

${\text{log}}_{2}\left(x\right)=-3$

$x={2}^{-3}=\frac{1}{8}$

${\text{log}}_{5}\left(x\right)=2$

${\mathrm{log}}_{3}\left(x\right)=3$

$x={3}^{3}=27$

${\text{log}}_{2}\left(x\right)=6$

${\text{log}}_{9}\left(x\right)=\frac{1}{2}$

$x={9}^{\frac{1}{2}}=3$

${\text{log}}_{18}\left(x\right)=2$

${\mathrm{log}}_{6}\left(x\right)=-3$

$x={6}^{-3}=\frac{1}{216}$

$\text{log}\left(x\right)=3$

$\text{ln}\left(x\right)=2$

$x={e}^{2}$

For the following exercises, use the definition of common and natural logarithms to simplify.

$\text{log}\left({100}^{8}\right)$

${10}^{\text{log}\left(32\right)}$

$32$

$2\text{log}\left(.0001\right)$

${e}^{\mathrm{ln}\left(1.06\right)}$

$1.06$

$\mathrm{ln}\left({e}^{-5.03}\right)$

${e}^{\mathrm{ln}\left(10.125\right)}+4$

$14.125$

## Numeric

For the following exercises, evaluate the base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ logarithmic expression without using a calculator.

${\text{log}}_{3}\left(\frac{1}{27}\right)$

${\text{log}}_{6}\left(\sqrt{6}\right)$

$\frac{1}{2}$

${\text{log}}_{2}\left(\frac{1}{8}\right)+4$

$6{\text{log}}_{8}\left(4\right)$

$4$

For the following exercises, evaluate the common logarithmic expression without using a calculator.

$\text{log}\left(10,000\right)$

$\text{log}\left(0.001\right)$

$-\text{3}$

$\text{log}\left(1\right)+7$

$2\text{log}\left({100}^{-3}\right)$

$-12$

For the following exercises, evaluate the natural logarithmic expression without using a calculator.

$\text{ln}\left({e}^{\frac{1}{3}}\right)$

$\text{ln}\left(1\right)$

$0$

$\text{ln}\left({e}^{-0.225}\right)-3$

$25\text{ln}\left({e}^{\frac{2}{5}}\right)$

$10$

## Technology

For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.

$\text{log}\left(0.04\right)$

$\text{ln}\left(15\right)$

$\text{2}.\text{7}0\text{8}$

$\text{ln}\left(\frac{4}{5}\right)$

$\text{log}\left(\sqrt{2}\right)$

$0.151$

$\text{ln}\left(\sqrt{2}\right)$

## Extensions

Is $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ in the domain of the function $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(x\right)?\text{\hspace{0.17em}}$ If so, what is the value of the function when $\text{\hspace{0.17em}}x=0?\text{\hspace{0.17em}}$ Verify the result.

No, the function has no defined value for $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ To verify, suppose $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ is in the domain of the function $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(x\right).\text{\hspace{0.17em}}$ Then there is some number $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}n=\mathrm{log}\left(0\right).\text{\hspace{0.17em}}$ Rewriting as an exponential equation gives: $\text{\hspace{0.17em}}{10}^{n}=0,$ which is impossible since no such real number $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ exists. Therefore, $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ is not the domain of the function $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(x\right).$

Is $\text{\hspace{0.17em}}f\left(x\right)=0\text{\hspace{0.17em}}$ in the range of the function $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(x\right)?\text{\hspace{0.17em}}$ If so, for what value of $\text{\hspace{0.17em}}x?\text{\hspace{0.17em}}$ Verify the result.

Is there a number $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}\mathrm{ln}x=2?\text{\hspace{0.17em}}$ If so, what is that number? Verify the result.

Yes. Suppose there exists a real number $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}\mathrm{ln}x=2.\text{\hspace{0.17em}}$ Rewriting as an exponential equation gives $\text{\hspace{0.17em}}x={e}^{2},$ which is a real number. To verify, let $\text{\hspace{0.17em}}x={e}^{2}.\text{\hspace{0.17em}}$ Then, by definition, $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right)=\mathrm{ln}\left({e}^{2}\right)=2.$

Is the following true: $\text{\hspace{0.17em}}\frac{{\mathrm{log}}_{3}\left(27\right)}{{\mathrm{log}}_{4}\left(\frac{1}{64}\right)}=-1?\text{\hspace{0.17em}}$ Verify the result.

Is the following true: $\text{\hspace{0.17em}}\frac{\mathrm{ln}\left({e}^{1.725}\right)}{\mathrm{ln}\left(1\right)}=1.725?\text{\hspace{0.17em}}$ Verify the result.

No; $\text{\hspace{0.17em}}\mathrm{ln}\left(1\right)=0,$ so $\text{\hspace{0.17em}}\frac{\mathrm{ln}\left({e}^{1.725}\right)}{\mathrm{ln}\left(1\right)}\text{\hspace{0.17em}}$ is undefined.

## Real-world applications

The exposure index $\text{\hspace{0.17em}}EI\text{\hspace{0.17em}}$ for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation $\text{\hspace{0.17em}}EI={\mathrm{log}}_{2}\left(\frac{{f}^{2}}{t}\right),$ where $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is the “f-stop” setting on the camera, and $t$ is the exposure time in seconds. Suppose the f-stop setting is $\text{\hspace{0.17em}}8\text{\hspace{0.17em}}$ and the desired exposure time is $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ seconds. What will the resulting exposure index be?

Refer to the previous exercise. Suppose the light meter on a camera indicates an $\text{\hspace{0.17em}}EI\text{\hspace{0.17em}}$ of $\text{\hspace{0.17em}}-2,$ and the desired exposure time is 16 seconds. What should the f-stop setting be?

$2$

The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula $\text{\hspace{0.17em}}\mathrm{log}\frac{{I}_{1}}{{I}_{2}}={M}_{1}-{M}_{2}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}M\text{\hspace{0.17em}}$ is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0. http://earthquake.usgs.gov/earthquakes/world/historical.php. Accessed 3/4/2014. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
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can get some help basic precalculus
What do you need help with?
Andrew
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ismail
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yeah, it does. why do we attempt to gain all of them one side or the other?
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whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
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foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
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give me an example of a problem so that I can practice answering
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dont forget the cube in each variable ;)
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of she solves that, well ... then she has a lot of computational force under her command ....
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explain this