Proof: Log of an Exponential

Let's prove the following theorem:

This theorem says that, if we take the log of an exponential, and the bases are the same, then the output is the exponent.

For example:

log₂(2³) = 3

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

2³ = 8

Using substitution:

log₂(2³) = log₂8

We also know that:

log₂8 = 3

Thus:

log₂(2³) = 3

Proof:

Proof Table

#	Claim	Reason
1	logx(xp) = logxz	if xp = z, then logx(xp) = logxz (Substitution)
2	logxz = p	if xp = z, then logxz = p (Log Definition)
3	logx(xp) = p	if logx(xp) = logxz and logxz = p, then logx(xp) = p (Transitive Property of Equality)

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