Difference between revisions of "Exponential and Logarithmic Equations"

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\therefore T &= 0.1985R^{1.5011}   
 
\therefore T &= 0.1985R^{1.5011}   
 
\end{align}</math>
 
\end{align}</math>
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== Change of Base Formula ==
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<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).
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Proof:<br>
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<math>\log_{y}x = b</math>
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<math>\ y^b = x</math>
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Put both sides to <math>\log_{a}</math>
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<math>\log_{a}y^b = \log_{a}x</math>
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<math>\ b\log_{a}y = \log_{a}x</math>
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<math>\ b = \frac{\log_{a}x}{\log_{a}y}</math>
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Replace <math>\ b</math> from first line
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<math>\log_{y}x = \frac{\log_{a}x}{\log_{a}y}</math>
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== Swap of Base and Exponent Formula ==
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<math>a^{\log_{b}c}=c^{\log_{b}a}</math> where a or c must not be equal to 1.
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Proof:
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<math> log_{a}b = \frac{1}{log_{b}a}</math> by the change of base formula above.
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Note that <math>a=c^{log_{c}a}</math>. Then
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<math>a^{log_{b}c}</math> can be rewritten as
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<math>({c^{log_{c}a}})^{ log_{b}c}</math> or by the exponential rule as
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<math>c^{{log_{c}a}*{log_{b}c}}</math>
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using the inverse rule noted above, this is equal to
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<math>c^{ {log_{c}a} * { \frac{1}{log_{c}b} } }</math>
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and by the change of base formula
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<math>c^{log_{b}a}</math>
  
 
==Resources==
 
==Resources==

Revision as of 12:38, 23 January 2022

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

Licensing

Content obtained and/or adapted from: