Difference between revisions of "Logarithmic Functions"

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In mathematics you can find the inverse of an exponential function by switching x and y around:  
 
In mathematics you can find the inverse of an exponential function by switching x and y around:  
 
<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>.
 
<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===
 
===Laws of Logarithmic Functions===
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A '''logarithmic equation''' is an equation wherein one or more of the terms is a logarithm.
 
A '''logarithmic equation''' is an equation wherein one or more of the terms is a logarithm.
  
e.g. Solve <math>\lg x + \lg (x+2) = 2</math> <ref group="note"><math>\lg</math> is another way of writing <math>\log_{10}</math></ref>
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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>).
  
 
<math>\begin{align}
 
<math>\begin{align}
\lg x + \lg (x+2) &= 2 \\
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\log x + \log (x+2) &= 2 \\
\lg (x(x+2)) &= 2 \\
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\log (x(x+2)) &= 2 \\
 
x(x+2) &= 100 \\
 
x(x+2) &= 100 \\
 
x^2 + 2x &= 100 \\
 
x^2 + 2x &= 100 \\
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* [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: