# 6.3 Logarithmic functions  (Page 6/9)

 Page 6 / 9

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

${15}^{b}=a$

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

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

${13}^{a}=142$

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

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

${e}^{n}=w$

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.

write down the polynomial function with root 1/3,2,-3 with solution
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
what is the answer to dividing negative index
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
give me the waec 2019 questions
the polar co-ordinate of the point (-1, -1)
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
tanh`(x-iy) =A+iB, find A and B
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
1+cos²A/cos²A=2cosec²A-1
test for convergence the series 1+x/2+2!/9x3