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.
  
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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}
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\lg x + \lg (x+2) &= 2 \\
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\log x + \log (x+2) &= 2 \\
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\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

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In mathematics you can find the inverse of an exponential function by switching x and y around: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = b^x\,} becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = b^y\,} . 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac { \log_b x}{ \log_b a}} . 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log x + \log (x+2) = 2} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log} is another way of writing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_{10}} ).

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \log x + \log (x+2) &= 2 \\ \log (x(x+2)) &= 2 \\ x(x+2) &= 100 \\ x^2 + 2x &= 100 \\ (x + 1)^2 &= 101 \\ x+1 &= \sqrt{101} \\ x &= -1 \pm \sqrt{101} \end{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_2 8 = \frac { \log 8}{ \log 2} = \frac {.9}{.3} = 3\,} now check Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^3 = 8\,}

Resources

Licensing

Content obtained and/or adapted from:

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