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← Solving Logarithmic Equations and Inequalities

Rule

Property of Inequality for Logarithms

Let

b

be a positive real number different than

1 .

The following statements hold true.

If b < 1 lo g_{b} x_{1} \geq lo g_{b} x_{2} ⇕ x_{1} \leq x_{2} If b > 1 lo g_{b} x_{1} \geq lo g_{b} x_{2} ⇕ x_{1} \geq x_{2}

These facts are also valid for strict inequalities.

Proof

The statements will be proved one at a time.

$b > 1$

If $b$ is greater than $1,$ the logarithmic function $f (x) = lo g_{b} x$ is increasing over its entire domain.

In the graph, it can be seen that

f (x_{1}) \geq f (x_{2})

if and only if

x_{1} \geq x_{2} .

Considering the definition of

f (x),

the statement is proven.

lo g_{b} x_{1} \geq lo g_{b} x_{2} \Leftrightarrow x_{1} \geq x_{2}

$b < 1$

If $b$ is greater than $0$ and less than $1,$ then $f (x) = lo g_{b} x$ is decreasing over its entire domain.