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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

How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku 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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