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