LOGARITHM ARGUMENTS OF LESS THAN OR EQUAL TO ZERO. IS IT POSSIBLE?

Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to ingrid.stewart@csn.edu.

What is the value of log

_{b}0?As we know, a logarithm can only be defined if the base

bis greater than 0.For a moment, let's assume that a logarithm argument equals 0, and let's log

_{b}0 be equal to some numbern,that is, log_{b}0 = n.In its exponential form this would be 0 = b

^{n}.You are challenged to find a base

band an exponentnfor which this is true!!! What about ifneuqals 0? But any numberbraised to a 0 power has a value of 1. What about ifnis a negative number? But any numberbraised to a negative number is a fraction.You will find that there is no such power

nthat would make an exponential expression be equal to 0 given a basebgreater than 0.

Therefore, log_{b}0 is not defined.What is the value of log

_{b}(a)?Logarithms of negative numbers are not defined in the real numbers. If you are expected to find the logarithm of a negative number, an answer of "undefined" is sufficient in most cases.

However, it is possible to evaluate a logarithm of negative numbers. The answer will be a complex number, a number of the form

a + bi, whereis the imaginary number equal to .iFor a moment, let's assume that the logarithm argument is a negative number, say a, and let's log

_{b}(a) be equal to some numbern, that is, log_{b}(a) = n. We will further assume that basebis not equal to the number.eNow let's use the

Change-of-Base Propertyto change log_{b}(a) to base.eNote that log

_{e}(a) is the the same thing as log_{e}(1 a). We can now use theProduct Ruleof logarithms to write the numerator as follows:log

_{e}(a) = log_{e}(1a)= log

_{e}(1) + log_{e}aand

Now the only problem is figuring out what log

_{e}(1) is equal to. It might look like an impossible thing to evaluate at first, but there is a pretty famous equation known asEuler's Identitythat can help us.

Euler's Identitystates:This result comes from Calculus and is a power series expansions of sine and cosine. We won't explain that too in-depth, but if you are interested, there is a nice page here which explains a bit more.

For now, let us simply take the natural logarithm of both sides of

Euler's Identity:Using th basic logarithm property

log, we find_{b}b^{x}=xSo now that we know what log

_{e}(1) is equal to, we can substitute it back into the equationThat is,

Now we have a formula for finding logarithms of negative numbers.

For example, if we were asked to find log

_{2}(10) and round the answer to three decimal places, we would get