If a > 0 (but not 1), a^x is strictly increasing or strictly decreasing, (hence passes the horizontal line test), and therefore has an inverse. We call the inverse of a^x - log_a(x) - the logarithmic function with base a .

Because the log is the inverse of the exponential, we have

• log_a(a^x) = x for every real x
• a ^{log _a x} = x, for every positive real x

Theorem:

1. log_a(xy) = log_ax + log_ay
2. log_a(x/y) = log_ax - log_ay
3. log_a(x^y) = y log_a x

The particular number a=e=2.71828... comes up so often that we use the shorthand notation log_a(x) = ln(x), the natural log, for this particular base.

We can convert between bases by

log_a x = ln (x) / ln(a)

Therefore, letting N=1,000,000 the sum of the first one million reciprocal integers is approximately ln(1,000,000)=13.815511! Try writing a computer program to calculuate 1 + 1/2 + 1/3 + 1/4 + ... + 1/N and see how close it gets...

Last modified Mon Oct 21 15:31:54 CDT 1996