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A math student, Andrea, asked,

i need help trying to figure out this problem....
write the equation 3^2=9 in logarithmic form
Somebody please HELP!

A logarithm is a function that is the inverse of an exponential function.  For example, if ea=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=eln a and b=eln b so ab=(eln a)(eln b), and therefore, ab=e(ln a+ln b).  Also, ab=(eln 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=eln 3.  So 32=(eln 3)2=e(ln 3)(2).  In other words the natural log of 32 is equal to 2 times the natural log of 3.

On 8/25/00 12:31:19 PM, Stephanie wrote:
I haven't taken algebra in over ten years. Can you
refresh my memory? I don't remember how you add,
subtract, multiply or divide negative exponents.

The most important rule of exponents is this:


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 xy)

(an)(am)=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/ab.

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 xy


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