##
Solving Logarithmic Equations

We want to find the solutions to

log(8

*x*) - log(1 +

) = 2.

Let us note that the equation is only defined when the input for
the logarithms is positive. Thus we need *x* > 0 and
1 + > 0.
Since the second condition is automatically satisfied when *x* > 0,
the equation is defined for all positive *x*.
We begin by combining the two logarithmic expressions into one
expression, using the rule that

Consequently our equation becomes

This is equivalent to saying that
= 10

^{2} = 100

Next we multiply both sides by the denominator on the left:

8

*x* = 100(1 +

),

or equivalently
8

*x* - 100 = 100

.

Next we square both sides to eliminate the square root term:

(8

*x* - 100)

^{2} = 10, 000

*x* 64

*x*^{2} - 1, 600

*x* + 10, 000 = 10, 000

*x*.

Simplifying we obtain the quadratic equation

8*x*^{2} - 1, 450*x* + 1, 250 = 0.

Using the quadratic formula, we compute two solutions to this
quadratic equation as

*x* =

.

Numerically the solutions equal
*x*_{1} 180.383 and
*x*_{2} 0.86621.

It is essential that you now check both answers in the original
equation!

For *x*_{1},

log(8

*x*_{1}) - log(1 +

)

2.000,

thus *x*_{1} is a solution to the equation.
On the other hand,

log(8

*x*_{2}) - log(1 +

)

.555,

thus *x*_{2} is **not** a solution to the equation.
**Answer:** The equation has exactly one solution, namely

*x* =

180.383.

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