Algebra Know-It-ALL

You can work out an example using the same numerical arguments as before. Again, follow
along with your calculator:

log 10 (3 /4) = log 10 3 − log 10 4

Working out both sides and approximating the results to four decimal places:

log 10 0.7500 ≈ 0.4771 − 0.6021

≈ −0.1250

Are you confused?

If you take the base-10 log of 0.7500 directly with your calculator and round it off to four decimal places,
you’ll see −0.1249, not −0.1250. What’s going on? Why is there a discrepancy between these two methods
of determining the base-10 log of 3/4? It’s not a flaw in the calculator, and it’s not your imagination. This
is an example of a phenomenon called rounding error, which often occurs when repeated calculations are
done using approximate values. This isn’t the last time you’re going to see it!

Changing a power to a product

Logarithms simplify the raising of a number to a power. This tactic is useful when the argu-
ment does not have two whole numbers, and it’s especially handy when irrationals enter the
scene. Let x be a positive real number, and let y be any real number. The base-b logarithm of
x raised to the power y can be rearranged as a product:

logbxy=y logbx

Again, this works for any positive base b. Here’s an example using the same arguments as
before, carried out to four decimal places:

log 10 (3^4 )= 4 log 10 3

log 10 81 ≈ 4.0000 × 0.4771

≈ 1.9084

Now let’s try an example in which both of the numbers in the input argument are decimals.
We’ll go to three places:

log 10 (2.6351.078)= 1.078 log 10 2.635

≈ 1.078 × 0.421

≈ 0.454

We’ll return to this result later.

How Logarithms Work 483