
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 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, e^{3}e^{4}=e^{7}. And (e^{2})^{3} is e^{6}. 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 ninesentence 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 3^{2}=9. 3=e^{ln 3}. So 3^{2}=(e^{ln 3})^{2}=e^{(ln 3)(2)}. In other words the natural log of 3^{2} is equal to 2 times the natural log of 3.
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:
(a^{n})(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 0^{0} is not defined, while others say it is 1. (More about this in Euler and x^{y})
(a^{n})(a^{m})=a^{(n+m)} means, for example, that 2^{3} times 2^{11} equals 2^{14}. But it also works for negative exponents. 2^{3} times 2^{2} equals 2^{1}. That is, 8 times 2^{2} is 2, so 2^{2} is 1/4, which is also 1/2^{2}. 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 b^{4}/a^{3}. This rule will help you simplify most negative exponents you run into.
Euler and Exponents  Euler's formula, and x^{y}
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