Proof: When the Exponent is a Logarithm

Let's prove the following theorem:

This theorem says that, if we have an exponential operation, and the exponent is a logarithm, and the bases are the same, then the output is the log input.

For example:

2^(log₂8) = 8

We can confirm this by evaluating the logarithm on the left. We know that:

log₂8 = 3

Using substitution:

2^(log₂8) = 2³

We also know that:

2³ = 8

Thus:

2^(log₂8) = 8

Proof:

Proof Table

#	Claim	Reason
1	logbx = p	if bp = x, then logbx = p (Log Definition)
2	b(logbx) = bp	if logbx = p, then b(logbx) = bp (Substitution)
3	b(logbx) = x	if b(logbx) = bp and bp = x, then b(logbx) = x (Transitive Property of Equality)

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