Editing Elementary algebra (section)

=== Exponential and logarithmic equations ===
{{Main|Logarithm}}
[[File:Binary logarithm plot with ticks.svg|right|thumb|upright=1.35|alt=Graph showing a logarithm curves, which crosses the ''x''-axis where ''x'' is 1 and extend towards minus infinity along the ''y''-axis.|The [[graph of a function|graph]] of the logarithm to base 2 crosses the [[x axis|''x'' axis]] (horizontal axis) at 1 and passes through the points with [[coordinate]]s {{nowrap|(2, 1)}}, {{nowrap|(4, 2)}}, and {{nowrap|(8, 3)}}. For example, {{nowrap|log<sub>2</sub>(8) {{=}} 3}}, because {{nowrap|2<sup>3</sup> {{=}} 8.}} The graph gets arbitrarily close to the ''y'' axis, but [[asymptotic|does not meet or intersect it]].]]

An exponential equation is one which has the form <math>a^x = b</math> for <math>a > 0</math>,<ref>Aven Choo, ''LMAN OL Additional Maths Revision Guide 3'', Publisher Pearson Education South Asia, 2007, {{ISBN|9810600011}}, 9789810600013, [https://books.google.com/books?id=NsBXDMrzcJIC&dq=%22+exponential+equation+%22+aX+%3D+b&pg=RA2-PA29 page 105]</ref> which has solution

: <math>x = \log_a b = \frac{\ln b}{\ln a}</math>

when <math>b > 0</math>. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if

: <math>3 \cdot 2^{x - 1} + 1 = 10</math>

then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain

: <math>2^{x - 1} = 3</math>

whence

: <math>x - 1 = \log_2 3</math>

or

: <math>x = \log_2 3 + 1.</math>

A logarithmic equation is an equation of the form <math>log_a(x) = b</math> for <math>a > 0</math>, which has solution

: <math>x = a^b.</math>

For example, if

: <math>4\log_5(x - 3) - 2 = 6</math>

then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get

: <math>\log_5(x - 3) = 2</math>

whence

: <math>x - 3 = 5^2 = 25</math>

from which we obtain

: <math>x = 28.</math>