Algebra Know-It-ALL

Changing a reciprocal to a negative

The logarithm (to any base b) of the reciprocal of a number is equal to the negative of the

logarithm of that number. Stated mathematically, if x is a positive real number, then

logb (1/x)=−(logbx)

This is a special case of division simplifying to subtraction. Now let’s look at a numerical

example. Suppose x= 3 (exactly) and we use natural logs, as follows:

ln (1/3) =−(ln 3)

Using a calculator, we can evaluate both expressions. This time, let’s go to nine decimal places.

ln 0.333333333 ≈−(ln 3.000000000)

−1.098612290≈−1.098612289

We have some rounding error here, but it’s only at the ninth decimal place, equivalent to one

part in 1,000,000,000!

Reciprocal within an exponent

What happens when we have a reciprocal in an exponent? Suppose x is a positive real number,

andy is any real number except zero. Then the logarithm (to any positive base b) of the yth

root of x (also denoted as x to the 1/y power) is equal to the log of x, divided by y:

logb (x1/y)= (logbx) / y

Let’s try this with x= 8 and y= 1/3, considering both values exact, and using natural logs

evaluated to five decimal places. We get

ln (81/3)= (ln 8) / 3

We know that the 1/3 power (or cube root) of 8 is equal to 2. Therefore

ln 2 = (ln 8) / 3

0.69315 ≈ 2.07944 / 3

0.69315 = 0.69315

The results agree to all five decimal places here, so we can write a plain equals sign instead of

a squiggly one.

Log conversions

Here are a couple of useful rules for converting natural logs to common logs and vice-versa.

You’ll sometimes have to do this, especially if you get into physics or engineering.

484 Logarithms and Exponentials