What is a logarithm?

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Chapter 3


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

If y = ax


logay = x.

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

For example,
25 = 64


log2 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 log10y."

103 = 1,000.

log101,000 = log 1,000 = 3.

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