Difference between revisions of "Exponential and Logarithmic Equations"
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+ | == Logarithms and Exponents == | ||
+ | |||
+ | A '''logarithm''' is the inverse function of an exponent. | ||
+ | |||
+ | e.g. The inverse of the function <math>f(x) = 3^x</math> is <math>f^{-1}(x) = \log_3 x</math>. | ||
+ | |||
+ | In general, <math>y = b^x \iff x = \log_b y</math>, given that <math>b > 0</math>. | ||
+ | |||
+ | == Laws of Logarithms == | ||
+ | |||
+ | The laws of logarithms can be derived from the laws of exponentiation: | ||
+ | |||
+ | <math>\begin{align} | ||
+ | x^{a+b} = x^a \times x^b &\iff \log a + \log b = \log ab \\ | ||
+ | x^{a-b} = x^a \div x^b &\iff \log a - \log b = \log a/b \\ | ||
+ | (x^a)^b = x^{ab} &\iff \log a^b = b \log a | ||
+ | \end{align}</math> | ||
+ | |||
+ | These laws apply to logarithms of any given base | ||
+ | |||
+ | == Natural Logarithms == | ||
+ | |||
+ | The '''natural logarithm''' is a logarithm with base <math>e</math>, where <math>e</math> is a constant such that the function <math>e^x</math> is its own derivative. | ||
+ | |||
+ | The natural logarithm has a special symbol: <math>\ln x</math> | ||
+ | |||
+ | The graph <math>y = e^{kx}</math> exhibits exponential growth when <math>k > 0</math> and exponential decay when <math>k < 0</math>. The inverse graph is <math>y = \frac{1}{k} \ln x</math>. [https://www.desmos.com/calculator/azbv4kz38z Here] is an interactive graph which shows the two functions as inverses of one another. | ||
+ | |||
+ | == Solving Logarithmic and Exponential Equations == | ||
+ | |||
+ | An '''exponential equation''' is an equation in which one or more of the terms is an exponential function. e.g. <math>5^x = 2^{x+2}</math>. Exponential equations can be solved with logarithms. | ||
+ | |||
+ | e.g. Solve <math>3^{x+1} = 4^{2x-1}</math> | ||
+ | |||
+ | <math>\begin{align} | ||
+ | 3^{x+1} &= 4^{2x-1} \\ | ||
+ | (x+1)\ln 3 &= (2x-1)\ln 4 \\ | ||
+ | x\ln 3 + \ln 3 &= 2x\ln 4 - \ln 4 \\ | ||
+ | \ln 3 + \ln 4 &= x(2\ln 4 - \ln 3) \\ | ||
+ | x &= \frac{\ln 3 + \ln 4}{2\ln 4 - \ln 3} \\ | ||
+ | x &\approx 1.4844 | ||
+ | \end{align}</math> | ||
+ | |||
+ | 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> | ||
+ | |||
+ | <math>\begin{align} | ||
+ | \lg x + \lg (x+2) &= 2 \\ | ||
+ | \lg (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}</math> | ||
+ | |||
+ | == Converting Relationships to a Linear Form == | ||
+ | |||
+ | In maths and science, it is easier to deal with linear relationships than non-linear relationships. Logarithms can be used to convert some non-linear relationships into linear relationships. | ||
+ | |||
+ | === Exponential Relationships === | ||
+ | |||
+ | An '''exponential relationship''' is of the form <math>y = ab^x</math>. If we take the natural logarithm of both sides, we get <math>\ln y = \ln a + x \ln b</math>. We now have a linear relationship between <math>\ln y</math> and <math>x</math>. | ||
+ | |||
+ | e.g. The following data is related with an exponential relationship. Determine this exponential relationship, then convert it to linear form. | ||
+ | |||
+ | {| style="width:30%" class="wikitable" | ||
+ | |- | ||
+ | ! x !! y | ||
+ | |- | ||
+ | | 0 || 5 | ||
+ | |- | ||
+ | | 2 || 45 | ||
+ | |- | ||
+ | | 4 || 405 | ||
+ | |} | ||
+ | |||
+ | <math>\begin{align} | ||
+ | \text{Exponential relationship } \implies y &= ab^x \\ | ||
+ | 5 &= ab^0 = a(1) \\ | ||
+ | a &= 5 \\ | ||
+ | y &= 5b^x \\ | ||
+ | 45 &= 5b^2 \\ | ||
+ | 9 &= b^2 \\ | ||
+ | b &= 3 \\ | ||
+ | y &= 5(3^x) | ||
+ | \end{align}</math> | ||
+ | |||
+ | Now convert it to linear form by taking the natural logarithm of both sides: | ||
+ | |||
+ | <math>\begin{align} | ||
+ | y &= 5(3^x) \\ | ||
+ | \ln y &= \ln 5 + x \ln 3 | ||
+ | \end{align}</math> | ||
+ | |||
+ | === Power Relationships === | ||
+ | |||
+ | A '''power relationship''' is of the form <math>y = ax^b</math>. If we take the natural logarithm of both sides, we get <math>\ln y = \ln a + b \ln x</math>. This is a linear relationship between <math>\ln y</math> and <math>\ln x</math>. | ||
+ | |||
+ | e.g. The amount of time that a planet takes to travel around the sun (its orbital period) and its distance from the sun are related by a power law. Use the following data to deduce this power law: | ||
+ | |||
+ | {| class="wikitable" | ||
+ | |- | ||
+ | ! Planet !! Distance from Sun /10<sup>6</sup> km !! Orbital Period /days | ||
+ | |- | ||
+ | | Earth || 149.6 || 365.2 | ||
+ | |- | ||
+ | | Mars || 227.9 || 687.0 | ||
+ | |- | ||
+ | | Jupiter || 778.6 || 4331 | ||
+ | |} | ||
+ | |||
+ | <math>\begin{align} | ||
+ | \text{Power law}\implies T &= aR^b \\ | ||
+ | \text{Use Earth data}\implies 365.2 &= a(149.6^b) \\ | ||
+ | \ln 365.2 &= \ln a + b \ln 149.6 \\ | ||
+ | \text{Use Mars data}\implies 687.0 &= a(227.9^b) \\ | ||
+ | \ln 687.0 &= \ln a + b \ln 227.9 \\ | ||
+ | \ln 687.0 - \ln 365.2 &= \ln a - \ln a + b\ln 227.9 - b\ln 149.6 \\ | ||
+ | \ln \frac{687.0}{365.2} &= 0 + b(\ln\frac{227.9}{149.6}) \\ | ||
+ | b &= \frac{\ln \tfrac{687.0}{365.2}}{\ln\tfrac{227.9}{149.6}} \approx 1.5011 \\ | ||
+ | \ln 365.2 &= \ln a + 1.5011 \ln 149.6 \\ | ||
+ | \ln a &= \ln 365.2 - \ln 1839.9 \\ | ||
+ | \ln a &= \ln 0.1985 \\ | ||
+ | a &= 0.1985 \\ | ||
+ | \therefore T &= 0.1985R^{1.5011} | ||
+ | \end{align}</math> | ||
+ | |||
+ | == Change of Base Formula == | ||
+ | <math>\log_{y}x=\frac{\log_{a}x} {\log_{a}y}</math> where ''a'' is any positive number, distinct from 1. Generally, ''a'' is either 10 (for common logs) or ''e'' (for natural logs). | ||
+ | |||
+ | Proof:<br> | ||
+ | <math>\log_{y}x = b</math> | ||
+ | |||
+ | <math>\ y^b = x</math> | ||
+ | |||
+ | Put both sides to <math>\log_{a}</math> | ||
+ | |||
+ | <math>\log_{a}y^b = \log_{a}x</math> | ||
+ | |||
+ | <math>\ b\log_{a}y = \log_{a}x</math> | ||
+ | |||
+ | <math>\ b = \frac{\log_{a}x}{\log_{a}y}</math> | ||
+ | |||
+ | Replace <math>\ b</math> from first line | ||
+ | |||
+ | <math>\log_{y}x = \frac{\log_{a}x}{\log_{a}y}</math> | ||
+ | |||
+ | == Swap of Base and Exponent Formula == | ||
+ | |||
+ | <math>a^{\log_{b}c}=c^{\log_{b}a}</math> where a or c must not be equal to 1. | ||
+ | |||
+ | Proof: | ||
+ | |||
+ | <math> log_{a}b = \frac{1}{log_{b}a}</math> by the change of base formula above. | ||
+ | |||
+ | Note that <math>a=c^{log_{c}a}</math>. Then | ||
+ | |||
+ | <math>a^{log_{b}c}</math> can be rewritten as | ||
+ | |||
+ | <math>({c^{log_{c}a}})^{ log_{b}c}</math> or by the exponential rule as | ||
+ | |||
+ | <math>c^{{log_{c}a}*{log_{b}c}}</math> | ||
+ | |||
+ | using the inverse rule noted above, this is equal to | ||
+ | |||
+ | <math>c^{ {log_{c}a} * { \frac{1}{log_{c}b} } }</math> | ||
+ | |||
+ | and by the change of base formula | ||
+ | |||
+ | <math>c^{log_{b}a}</math> | ||
+ | |||
==Resources== | ==Resources== | ||
+ | * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Logarithmic%20and%20Exponential%20Equations/Esparza%201093%20Notes%207.4.pdf Logarithmic and Exponential Equations]. Written notes created by Professor Esparza, UTSA. | ||
* [https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:exp-eq-log/a/solving-exponential-equations-with-logarithms Solving Exponential Equations with Logarithms], Khan Academy | * [https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:exp-eq-log/a/solving-exponential-equations-with-logarithms Solving Exponential Equations with Logarithms], Khan Academy | ||
* [https://www.mcckc.edu/tutoring/docs/br/math/expon_logar/Solving_Exponential_and_Logarithmic_Equations.pdf Solving Exponential and Logarithmic Equations], Metropolitan Community College Kansas City | * [https://www.mcckc.edu/tutoring/docs/br/math/expon_logar/Solving_Exponential_and_Logarithmic_Equations.pdf Solving Exponential and Logarithmic Equations], Metropolitan Community College Kansas City | ||
* [https://tutorial.math.lamar.edu/classes/calci/explogeqns.aspx Exponential and Logarithmic Equations], Paul's Online Notes | * [https://tutorial.math.lamar.edu/classes/calci/explogeqns.aspx Exponential and Logarithmic Equations], Paul's Online Notes | ||
+ | |||
+ | == Licensing == | ||
+ | Content obtained and/or adapted from: | ||
+ | * [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Logarithmic_and_Exponential_Functions Logarithmic and Exponential Functions, Wikibooks: A-level Mathematics] under a CC BY-SA license | ||
+ | * [https://en.wikibooks.org/wiki/Algebra/Logarithms Logarithms, Wikibooks: Algebra] under a CC BY-SA license |
Latest revision as of 12:40, 23 January 2022
Contents
Logarithms and Exponents
A logarithm is the inverse function of an exponent.
e.g. The inverse of the function is .
In general, , given that .
Laws of Logarithms
The laws of logarithms can be derived from the laws of exponentiation:
These laws apply to logarithms of any given base
Natural Logarithms
The natural logarithm is a logarithm with base , where is a constant such that the function is its own derivative.
The natural logarithm has a special symbol:
The graph exhibits exponential growth when and exponential decay when . The inverse graph is . Here is an interactive graph which shows the two functions as inverses of one another.
Solving Logarithmic and Exponential Equations
An exponential equation is an equation in which one or more of the terms is an exponential function. e.g. . Exponential equations can be solved with logarithms.
e.g. Solve
A logarithmic equation is an equation wherein one or more of the terms is a logarithm.
e.g. Solve
Converting Relationships to a Linear Form
In maths and science, it is easier to deal with linear relationships than non-linear relationships. Logarithms can be used to convert some non-linear relationships into linear relationships.
Exponential Relationships
An exponential relationship is of the form . If we take the natural logarithm of both sides, we get . We now have a linear relationship between and .
e.g. The following data is related with an exponential relationship. Determine this exponential relationship, then convert it to linear form.
x | y |
---|---|
0 | 5 |
2 | 45 |
4 | 405 |
Now convert it to linear form by taking the natural logarithm of both sides:
Power Relationships
A power relationship is of the form . If we take the natural logarithm of both sides, we get . This is a linear relationship between and .
e.g. The amount of time that a planet takes to travel around the sun (its orbital period) and its distance from the sun are related by a power law. Use the following data to deduce this power law:
Planet | Distance from Sun /106 km | Orbital Period /days |
---|---|---|
Earth | 149.6 | 365.2 |
Mars | 227.9 | 687.0 |
Jupiter | 778.6 | 4331 |
Change of Base Formula
where a is any positive number, distinct from 1. Generally, a is either 10 (for common logs) or e (for natural logs).
Proof:
Put both sides to
Replace from first line
Swap of Base and Exponent Formula
where a or c must not be equal to 1.
Proof:
by the change of base formula above.
Note that . Then
can be rewritten as
or by the exponential rule as
using the inverse rule noted above, this is equal to
and by the change of base formula
Resources
- Logarithmic and Exponential Equations. Written notes created by Professor Esparza, UTSA.
- Solving Exponential Equations with Logarithms, Khan Academy
- Solving Exponential and Logarithmic Equations, Metropolitan Community College Kansas City
- Exponential and Logarithmic Equations, Paul's Online Notes
Licensing
Content obtained and/or adapted from:
- Logarithmic and Exponential Functions, Wikibooks: A-level Mathematics under a CC BY-SA license
- Logarithms, Wikibooks: Algebra under a CC BY-SA license