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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

what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
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
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Practice Key Terms 3

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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