6.3 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.

what are you up to?
nothing up todat yet
Miranda
hi
jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
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Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
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Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
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Nokwanda
C'est comment
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Hi
Amanda
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SORIE
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Ranjay
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ANSHU
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Chinni
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Chinni
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Hassan
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SORIE
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Abdel
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SORIE
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function