This theorem allows us to convert a log of a product to a sum of the log of each operand.

For example:

log₂(8 ⋅ 16) = (log₂8) + (log₂16)

log₂(8 ⋅ 16) = 3 + 4

log₂(8 ⋅ 16) = 7

Before calculators were invented, multiplying large numbers would take a long time. People used the Log Product Rule to convert multiplication problems to addition problems which are easier to solve.

Let's try multiplying two numbers using the following algorithm:

1. Calculate log base 10 of both operands using a log table.
2. Add the results.
3. Calculate 10 to the power of the result from step 2 using an "anti-log" table.

For example, suppose we want to calculate 14⋅18.

First, we calculate:

log₁₀(14 ⋅ 18)

This is equal to

(log₁₀14) + (log₁₀18)

Then, the log table tells us that:

log₁₀14 = 1.146

log₁₀18 = 1.255

Thus:

(log₁₀14) + (log₁₀18) = 2.401

Finally, the anti-log table tells us that:

10^2.401 = 251.77

14⋅18 = 252, so we came quite close to the answer.

The key to proving this theorem is the following exponentiation property:

x^m ⋅ xⁿ = x^(m+n)

Quiz (1 point)

• (x^m) ⋅ (xⁿ) = x^{(m + n)}

• if log_xz = p, then x^p = z

• if log_xz = p, then x^p = z

• if log_xz = p, then x^p = z

• if the following are true:

  • a = x
  • b = y
  • c = x ⋅ y

  then c = a ⋅ b

• if the following are true:

  • a = b
  • b = c

  then a = c

• if b^m = bⁿ, then m = n

• if the following are true:

  • a = x
  • b = y
  • c = z
  • z = x + y

  then c = a + b

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.