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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.

log button on calculator

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 x to mean log10 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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