Difference between revisions of "Logarithmic Functions"

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==Logarithmic Functions==
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[[Image:Logexponential.svg|right|400px]]
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In mathematics you can find the inverse of an exponential function by switching x and y around:
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<math>y = b^x\,</math> becomes <math>x = b^y\,</math>. The problem arises on how to find the value  of y. The logarithmic function solved this problem. All conversions of logarithmic function into an exponential function follow the same pattern: <math>x = b^y\,</math> becomes <math>y = \log_b x\,</math>. If a log is given without a written b then b=10. Also with logarithmic functions, b > 0 and <math>b \ne 1</math>. There are 2 cases where the log is equal to x: <math>\log_bb^X = X\,</math> and <math>b^{\log_bX} = X\,</math>.
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To recap, a '''logarithm''' is the inverse function of an exponent.
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e.g. The inverse of the function <math>f(x) = 3^x</math> is <math>f^{-1}(x) = \log_3 x</math>.
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In general, <math>y = b^x \iff x = \log_b y</math>, given that <math>b > 0</math>.
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===Laws of Logarithmic Functions===
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When X and Y are positive.
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* <math>\log_bXY = \log_bX + \log_bY\,</math>
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* <math>\log_b \frac{X}{Y} = \log_bX - \log_bY\,</math>
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* <math>\log_b X^k = k \log_bX\,</math>
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===Change of Base===
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When x and b are positive real numbers and are not equal to 1. Then you can write <math>\log_a x\,</math> as <math>\frac { \log_b x}{ \log_b a}</math>. This works for the natural log as well. here is an example:
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===Solving a Logarithmic Equation===
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A '''logarithmic equation''' is an equation wherein one or more of the terms is a logarithm.
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e.g. Solve <math>\log x + \log (x+2) = 2</math> (<math>\log</math> is another way of writing <math>\log_{10}</math>).
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<math>\begin{align}
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\log x + \log (x+2) &= 2 \\
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\log (x(x+2)) &= 2 \\
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x(x+2) &= 100 \\
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x^2 + 2x &= 100 \\
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(x + 1)^2 &= 101 \\
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x+1 &= \sqrt{101} \\
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x &= -1 \pm \sqrt{101}
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\end{align}</math>
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<math>\log_2 8 = \frac { \log 8}{ \log 2} = \frac {.9}{.3} = 3\,</math> now check <math>2^3 = 8\,</math>
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==Resources==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Logarithmic_Functions/MAT1053_M5.2Logarithmic_Functions.pdf Logarithmic Functions], Book Chapter
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Logarithmic_Functions/MAT1053_M5.2Logarithmic_Functions.pdf Logarithmic Functions], Book Chapter
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Logarithmic_Functions/MAT1053_M5.2Logarithmic_FunctionsGN.pdf Guided Notes]
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Logarithmic_Functions/MAT1053_M5.2Logarithmic_FunctionsGN.pdf Guided Notes]
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== Licensing ==
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Content obtained and/or adapted from:
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* [https://en.wikibooks.org/wiki/A-level_Mathematics/OCR/C2/Logarithms_and_Exponentials Logarithms and Exponentials, Wikibooks: A-level Mathematics/OCR/C2] under a CC BY-SA license

Latest revision as of 14:47, 21 October 2021

Logarithmic Functions

Logexponential.svg

In mathematics you can find the inverse of an exponential function by switching x and y around: becomes . The problem arises on how to find the value of y. The logarithmic function solved this problem. All conversions of logarithmic function into an exponential function follow the same pattern: becomes . If a log is given without a written b then b=10. Also with logarithmic functions, b > 0 and . There are 2 cases where the log is equal to x: and .

To recap, a logarithm is the inverse function of an exponent.

e.g. The inverse of the function is .

In general, , given that .


Laws of Logarithmic Functions

When X and Y are positive.

Change of Base

When x and b are positive real numbers and are not equal to 1. Then you can write as . This works for the natural log as well. here is an example:

Solving a Logarithmic Equation

A logarithmic equation is an equation wherein one or more of the terms is a logarithm.

e.g. Solve ( is another way of writing ).

now check

Resources

Licensing

Content obtained and/or adapted from: