A math student, Andrea, asked,

A logarithm is a function that is the inverse of an exponential function.  For example, if e^a=b then a is the natural (that is, base e) logarithm of b.  In shorthand, a=ln b.  The beauty of logarithms is that they magically turn multiplication into addition, and powers into multiplication.  That's because of the way exponents work.  For example, e³e⁴=e⁷.  And (e²)³ is e⁶.  Expanding on this concept just a little bit, a=e^ln a and b=e^ln b so ab=(e^ln a)(e^ln b), and therefore, ab=e^{(ln a+ln b)}.  Also, a^b=(e^ln a)^b, which is the same as e^(ln a)(b).

Reread that paragraph a couple times.  It's a nine-sentence recap of just about all you need to know about logarithms.  Now, let's use this info to answer your specific question: How to use logarithms to explain 3²=9.  3=e^ln 3.  So 3²=(e^ln 3)²=e^{(ln 3)(2)}.  In other words the natural log of 3² is equal to 2 times the natural log of 3.

The most important rule of exponents is this:

(aⁿ)(a^m)=a^(n+m)

That rule applies whenever a is a real number, and n and m are integers, except 0^n is not defined if n is negative. Some people say 0⁰ is not defined, while others say it is 1. (More about this in Euler and x^y)

(aⁿ)(a^m)=a^(n+m) means, for example, that 2³ times 2¹¹ equals 2¹⁴. But it also works for negative exponents. 2³ times 2^-2 equals 2¹. That is, 8 times 2^-2 is 2, so 2^-2 is 1/4, which is also 1/2². In general, a^-b is 1/a^b.

This means whenever negative exponents appear in a fraction they can be moved to the other side of the fraction and have the sign of their exponent flipped. For example,

a^-3/b^-4 is equal to b⁴/a³. This rule will help you simplify most negative exponents you run into.

Related Pages in this website

Special Angles

d/dx (sin x) = cos x

Hyperbolic Functions

Euler and Exponents -- Euler's formula, and x^y

The webmaster and author of the Math Help site is Graeme McRae.
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