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Rule

Inverse Properties of Logarithms

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

log_b b^x =x and b^(log_b x)=x

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

rcr log 10^x=x & and & 10^(log x)=x [0.8em] ln e^x=x & and& e^(ln x)=x

Proof

The general equations will be proved one at a time.

log_b b^x =x

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

log_b b^x

log_b(a^m)= m* log_b(a)

x log_b b

log_b(b) = 1

x(1)

IdPropMult

Identity Property of Multiplication

x ✓

The logarithm of b^x with base b is equal to x.

b^(log_b x)=x

Let log_b x=a. Therefore, by the definition of a logarithm, b^a=x. log_b x=a ⇔ b^a=x This will be used to prove the identity.

b^(log_b x)

Substitute

log_b x= a

b^a

Substitute

b^a= x

x ✓

Therefore, b to the power of log_bx is equal to x.

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