Logarithmic functions are inverse functions of exponential equations.An equation b = a

^{x}can be written as a logarithmic equation as log_{a}b = x. As logarithms are used to solve an exponential equations, exponents are used to solve a logarithmic equation.## Solving a logarithmic equation Example

log (3x+1) = 2

The base of common logarithms is 10.

3x + 1 = 10

^{2}The equation is written in exponential form.

3x + 1 = 100

3x = 99 $\rightarrow $ x = 33

## Properties of logarithms in solving a logarithmic equation

**Example:**

log (7x+1) = log (x-2) +1

log (7x+1) - log (x-2) = 1 The logarithmic terms are grouped

$log\frac{7x+1}{x-2} =1$ Quotient rule for logarithms

$\frac{7x+1}{x-2} =10^{1}$ The equation is written in exponent form

$7x+1 = 10(x-2)$

$7x+1 = 10x -20$

$-3x = -21$ $\rightarrow $ $x = 7$

## Extraneous solutions for a logarithmic equation

**Example:**

log (x+2) + log(x-2) = 0

log [(x+2)(x-2)] = 0 Product rule

log ( x

^{2}- 4) = 0

x

^{2}- 4 = 10

^{0}Exponential form

x

^{2}- 4 = 1

x

^{2}= 5 $\rightarrow $ $x = \pm 5$

But the value x =-5 will render log(x-2) undefined. Hence x =-5 is an extraneous solutions and needs to be discarded.

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## Graphical Method

Consider the equation log (5-2x) = 0

To solve the equation graphically the x intercept of the graph y = 5-2x is to be found.The graph cuts the x axis at x =2.

The algebraical method also leads to the same solution.