So what is a logarithm? Let’s first see the notation, then I will explain. When taking the log (short for logarithm which I will use from now on) of a number, you need to know what base is being used. The notation for the log of *x* is log_{a}*x*. The *a* is the base and is usually a specified number. so examples using this notation are log_{2} 10, log_{10 }25, log_{18} 145, log_{e} 7.34. Let’s look at these.

log_{2} 10 is asking the question “What number can I use as the exponent of 2 so that the answer is 10?”. It turns out that 2^{3.321928094887} = 10 so log_{2} 10 = 3.321928094887.

log_{10 }25 is asking the question “What number can I use as the exponent of 10 so that the answer is 25?”. Well, 10^{1.39794} =25 so log_{10 }25 = 1.39794.

Are you getting the picture? What about log_{18} 145? This is asking the question “What number can I use as the exponent of 18 so that the answer is 145?”. 18^{1.72183} = 145 so log_{18} 145 = 1.72183.

Now let’s look at log_{e} 7.34. This shows that the base or the number we are taking the log of does not have to be an integer. The number *e*, which I have talked about before, is an irrational number, but it still can be used as a base. In fact, it is probably the most used base. Since *e*^{1.99334} = 7.34, then it follows that log_{e} 7.34 = 1.99334.

By the way, on most calculators, the log or log *x* key assumes that the base is 10. On most calculators as well, ln *x* means log_{e} *x*. “ln” means “natural log”.

Now log_{a}*x* and *a ^{x}* are inverses of each other. This means that one undoes the other. So if on your calculator, you find ln 7, then take that number and hit the

*e*key, you get the original 7 back. This works in reverse as well: Find

^{x}*e*

^{7}on your calculator, then hit the ln

*x*key. You will again get the 7 back.

In notation-speak, this inverseness is shown as

\[\begin{array}{l}

{{a}^{{\log}_{a}x}\hspace{0.33em}{=}\hspace{0.33em}{x}}\\

{{\log}_{a}{a}^{x}{=}\hspace{0.33em}{x}}

\end{array}

\]

In my next post, I will show how logarithms can be used to solve equations.