We can use this rule to change the base of the logarithm. Before calculators were invented, people used log tables to calculate the log of a value. But the log tables only contained answers for certain bases (usually 10). Thus, people used this rule to convert the log operation to the right base. Even now, many calculators only know how to compute logs in base 10 and e (natural log), so we still have to convert log operations to one of these two bases.

For example, suppose that we want to calculate log28 .

Our calculator does not know how to compute base 2, so we convert the expression as follows:

log28 = (log108) / (log102)

We can now use the calculator to evaluate log108 and log102. Or we can simplify the right side by substituting 23 for 8 and using the "Log of an Exponential" rule. Then:

log28 = (log10(23)) / (log102)

log28 = (3 ⋅ (log102)) / (log102)

log28 = 3

We can also go the other way. Suppose that we are trying to compute

(log1049) / (log107)

Suppose that we don't have a calculator, so we cannot evaluate either log1049 or log107 , but we can convert this division problem to:

log749

And we know that log749 is 2. Therefore,

(log1049) / (log107) = 2

High-Level Proof

Suppose that:

Then:

Thus:

ci = bk

Also:

(cj)k = bk

Which means:

ci = cj ⋅ k

Thus:

i = j ⋅ k

Which we can transform to:

k = i / j

Quiz (1 point)

Given that:
logcx = i
logcb = j
logbx = k
not (j = 0)

Prove that:
logbx = (logcx) / (logcb)

The following properties may be helpful:

Please write your proof in the table below. Each row should contain one claim. The last claim is the statement that you are trying to prove.

Step Claim Reason (optional) Error Message (if any)
1
2
3
4
5
6
7
8
9
10

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