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Rule

Inverse Properties of Logarithms

A logarithm and a power with the same base undo each other.

$lo g_{b} b^{x} = x and b^{l o g_{b} x} = x$

In particular, the above equations also hold true for common and natural logarithms.

lo g 1 0^{x} = x ln e^{x} = x and and 1 0^{l o g x} = x e^{l n x} = x

Proof

The general equations will be proved one at a time.

$lo g_{b} b^{x} = x$

This identity can be proved by using the Power Property of Logarithms and the definition of a logarithm.

lo g_{b} b^{x}

$lo g_{b} (a^{m}) = m \cdot lo g_{b} (a)$

x lo g_{b} b

$lo g_{b} (b) = 1$

x (1)

IdPropMult

Identity Property of Multiplication

x ✓

The logarithm of

b^{x}

with base

b

is equal to

x .

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

Let

lo g_{b} x = a .

Therefore, by the definition of a logarithm,

b^{a} = x .

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

This will be used to prove the identity.

b^{l o g_{b} x}

Substitute

$lo g_{b} x = a$

b^{a}

Substitute

$b^{a} = x$

x ✓

Therefore,

b

to the power of

lo g_{b} x

is equal to

x .

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Proof

logb​bx=x

blogb​x=x

$lo g_{b} b^{x} = x$

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