Exponential and Logarithmic Equations
Contents
Logarithms and Exponents
A logarithm is the inverse function of an exponent.
e.g. The inverse of the function is .
In general, 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 \iff x = \log_b y} , given that 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 b > 0} .
Laws of Logarithms
The laws of logarithms can be derived from the laws of exponentiation:
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} 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}}
These laws apply to logarithms of any given base
Natural Logarithms
The natural logarithm is a logarithm with base 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 e} , where 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 e} is a constant such that the function 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 e^x} is its own derivative.
The natural logarithm has a special symbol: 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 \ln x}
The graph 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 = e^{kx}} exhibits exponential growth when 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 k > 0} and exponential decay when 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 k < 0} . The inverse graph is 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 = \frac{1}{k} \ln x} . 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. 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 5^x = 2^{x+2}} . Exponential equations can be solved with logarithms.
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 3^{x+1} = 4^{2x-1}}
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} 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}}
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 \lg x + \lg (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 \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}}
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 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 = ab^x} . If we take the natural logarithm of both sides, we get 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 \ln y = \ln a + x \ln b} . We now have a linear relationship between 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 \ln y} and 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} .
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 |
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} \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}}
Now convert it to linear form by taking the natural logarithm of both sides:
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} y &= 5(3^x) \\ \ln y &= \ln 5 + x \ln 3 \end{align}}
Power Relationships
A power relationship is of the form 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 = ax^b} . If we take the natural logarithm of both sides, we get 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 \ln y = \ln a + b \ln x} . This is a linear relationship between 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 \ln y} and 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 \ln x} .
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 |
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} \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}}
Change of Base Formula
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_{y}x=\frac{\log_{a}x} {\log_{a}y}} where a is any positive number, distinct from 1. Generally, a is either 10 (for common logs) or e (for natural logs).
Proof:
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_{y}x = b}
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}
Put both sides to 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_{a}}
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_{a}y^b = \log_{a}x}
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 \ b\log_{a}y = \log_{a}x}
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 \ b = \frac{\log_{a}x}{\log_{a}y}}
Replace 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 \ b} from first line
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_{y}x = \frac{\log_{a}x}{\log_{a}y}}
Swap of Base and Exponent Formula
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 a^{\log_{b}c}=c^{\log_{b}a}} where a or c must not be equal to 1.
Proof:
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_{a}b = \frac{1}{log_{b}a}} by the change of base formula above.
Note that 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 a=c^{log_{c}a}} . Then
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 a^{log_{b}c}} can be rewritten as
or by the exponential rule 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 c^{{log_{c}a}*{log_{b}c}}}
using the inverse rule noted above, this is equal to
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 c^{ {log_{c}a} * { \frac{1}{log_{c}b} } }}
and by the change of base formula
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 c^{log_{b}a}}
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
