**Annotation category:**

Chapter 3

Note: |

A comfort with exponents is a prerequisite for this lesson. If you don't have that yet, review POWERS OF TEN.

If y = a^{x}

then

log_{a}y = x.

where we call x the **exponent**, and a the **base.**

For example,

2^{5} = 64

so

log_{2} 64 = 5.

In science, working with logarithms often helps to simplify and elucidate a problem.

1. *Ease in calculations:*
Logarithms obey certain algebraic rules - a property that makes them faster
and easier to manipulate than their linear counterparts. Before the age of
calculators and computers, logarithms were particularly important to
scientists; likewise those machines now rely on them!

log(a x b) = log(a) + log(b)

log(a/b) = log(a) - log(b)

2. *Ease in graphing:*
When the range of numbers to be plotted is very large, extending over
two or more orders of magnitude, it is usually better to plot the
logarthim (base 10) of the measured values rather than the values
themselves.

As logs make algebra neater, they can make plotted data look "neater" as well, thus facilitating the discernment of patterns (which a linear plot might hide.)

Since powers of ten are the basis for scientific notation, we usually work in
logarithms of base 10. That is, "a" (defined above) = 10. And

**log y** is
shorthand for **log _{10}y**."

So

10^{3} = 1,000.

log_{10}1,000
= log 1,000
= 3.