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log 15 ( a ) = b

15 b = a

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log 13 ( 142 ) = a

13 a = 142

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

n 4 = 103

log n ( 103 ) = 4

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y x = 39 100

log y ( 39 100 ) = x

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For the following exercises, solve for x by converting the logarithmic equation to exponential form.

log 2 ( x ) = 3

x = 2 3 = 1 8

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log 3 ( x ) = 3

x = 3 3 = 27

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log 9 ( x ) = 1 2

x = 9 1 2 = 3

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log 6 ( x ) = 3

x = 6 3 = 1 216

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

e ln ( 10.125 ) + 4

14.125

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Numeric

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

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

2 log ( 100 3 )

12

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

ln ( e 0.225 ) 3

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Technology

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

Extensions

Is x = 0 in the domain of the function f ( x ) = log ( x ) ? If so, what is the value of the function when x = 0 ? Verify the result.

No, the function has no defined value for x = 0. To verify, suppose x = 0 is in the domain of the function f ( x ) = log ( x ) . Then there is some number n such that n = log ( 0 ) . Rewriting as an exponential equation gives: 10 n = 0 , which is impossible since no such real number n exists. Therefore, x = 0 is not the domain of the function f ( x ) = log ( x ) .

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Is f ( x ) = 0 in the range of the function f ( x ) = log ( x ) ? If so, for what value of x ? Verify the result.

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Is there a number x such that ln x = 2 ? If so, what is that number? Verify the result.

Yes. Suppose there exists a real number x such that ln x = 2. Rewriting as an exponential equation gives x = e 2 , which is a real number. To verify, let x = e 2 . Then, by definition, ln ( x ) = ln ( e 2 ) = 2.

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Is the following true: log 3 ( 27 ) log 4 ( 1 64 ) = −1 ? Verify the result.

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Is the following true: ln ( e 1.725 ) ln ( 1 ) = 1.725 ? Verify the result.

No; ln ( 1 ) = 0 , so ln ( e 1.725 ) ln ( 1 ) is undefined.

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Real-world applications

The exposure index E I for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation E I = log 2 ( f 2 t ) , where f is the “f-stop” setting on the camera, and t is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?

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Refer to the previous exercise. Suppose the light meter on a camera indicates an E I of 2 , and the desired exposure time is 16 seconds. What should the f-stop setting be?

2

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The intensity levels I of two earthquakes measured on a seismograph can be compared by the formula log I 1 I 2 = M 1 M 2 where M 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.

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Questions & Answers

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
Practice Key Terms 3

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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