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← Solving Logarithmic Equations and Inequalities

Rule

Property of Equality for Logarithmic Equations

Let

b

be a positive real number different than

1 .

Two logarithms with the same base

b

are equal if and only if their arguments are equal.

$lo g_{b} x = lo g_{b} y \Leftrightarrow x = y$

Since logarithms are defined for positive numbers,

x

and

y

must be positive.

Proof

The biconditional statement will be proved in two parts.

$lo g_{b} x = lo g_{b} y \Rightarrow x = y$

Let

a

be a real number such that

a = lo g_{b} x .

lo g_{b} x = lo g_{b} y \to a = lo g_{b} y

By using the definition of a logarithm, the logarithm equation can be written as an exponential equation.

a = lo g_{b} y \Leftrightarrow b^{a} = y

Finally,

lo g_{b} x

can be substituted for

a

and the Inverse Property of Logarithms can be used.

b^{a} = y

$a = l o g_{b} x$

b^{l o g_{b} x} = y

$b^{l o g_{b} (m)} = m$

x = y ✓

$x = y \Rightarrow lo g_{b} x = lo g_{b} y$

The Inverse Property of Logarithms can be used to express

x

as

b^{l o g_{b} x} .

x = y \Leftrightarrow b^{l o g_{b} x} = y

Now, let

c

be a real number such that

c = lo g_{b} x .

b^{l o g_{b} x} = y \to b^{c} = y

Next, the definition of a logarithm can be used to rewrite the obtained exponential equation as a logarithmic equation.

b^{c} = y \Leftrightarrow c = lo g_{b} y

Finally, by its own defitnion,

c

is equal to

lo g_{b} x .

c = lo g_{b} y \Leftrightarrow l o g_{b} x = lo g_{b} y ✓

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