# 4. Logarithms to Base 10

by M. Bourne

Logarithms to Base 10 were used extensively for calculation up
until the calculator was adopted in the 1970s and 80s. The **concept** of
logarithms is still very important in many fields of science and
engineering. One example is acoustics.

Our calculators allow us to use logarithms to base 10. These
are called **common logarithms** ("log" on a calculator). We
normally do not include the 10 when we write logarithms to base
10.

We write

log

xto mean log_{10}x.

[This is the convention used on calculators, so most math text books follow along. Note, however, that "log" in computer programming generally means "log base *e*", which we learn about on the next page.]

### Examples

1. Find the logarithm of `5 623` to base `10`. Write this in exponential form.

Answer

Using our calculator, we have:

`log 5623 = 3.74997`

This means `10^3.74997 = 5623`

**Mental check:** `10^3 = 1,000` and `10^4 = 10,000`.

Our number `5623` is between these values, so it's at least reasonable.

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2. Find the antilogarithm of `-6.9788`.

Answer

This means "if `log N = -6.9788`, what is `N`?"

Using the logarithm laws, `N = 10^-6.9788 = 0.000 000 105`.

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