Before we begin, remember that the argument of a logarithm has to be greater than

0 .

With that in mind, let's start by excluding any values of

x

that would result in taking the logarithm of a number less than or equal to zero.

lo g_{4} (2 x + 5) \leq lo g_{4} (4 x - 3)

After we determine the range of values on which the inequality is defined, we can solve the inequality using the Properties of Logarithms and combine our solution sets.

Determining Where the Inequality is Defined

For the first logarithm to be defined,

2 x + 5

needs to be greater than zero.

2 x + 5 > 0

SubIneq

$LHS - 5 > RHS - 5$

2 x > - 5

DivIneq

$LHS / 2 > RHS / 2$

x > - \frac{5}{2}

Similarly, for the second logarithm to be defined,

4 x - 3

needs to be greater than zero.

4 x - 3 > 0

AddIneq

$LHS + 3 > RHS + 3$

4 x > 3

DivIneq

$LHS / 4 > RHS / 4$

x > \frac{3}{4}

Let's combine the solutions above to determine the interval where the inequality is defined.

The intersection of the above intervals, which is the interval where the inequality is defined, is $x > \frac{3}{4} .$

Solving the Inequality

To solve an inequality with logarithms with the same base on each side, we use the Inequality Property of Logarithmic Functions.

Inequality Property of Logarithmic Functions
$If b > 1, lo g_{b} x > lo g_{b} y if and only if x > y . If b > 1, lo g_{b} x < lo g_{b} y if and only if x < y .$

Exercise

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