1 Logarithms to base 10

Logarithms are very closely related to powers and can have any base number. However, one of the most commonly used was the logarithm to base 10, also known as the common logarithm. The process of taking a log to base 10, is the inverse (opposite operation) of raising the base 10 to a power.

In the example 10³ = 1000, 3 is the index or the power to which the number 10 is raised to give 1000.

When you take the logarithm, to base 10, of 1000 the answer is 3.

This is written as:

Log₁₀ (1000) = 3

Hence, the logarithm is the index to which the base was raised. Or: log₁₀ 10ⁿ = n

So, 10³ = 1000 and log₁₀ (1000) = 3 express the same fact but the latter is in the language of logarithms.

This idea can be generally defined as follows:

Here are some more examples that provide some general logarithm rules.

Since 2 = 2¹, then log₂ (2) = 1

Similarly, log₃ (3) = 1, and log₁₀ (10) = 1, and as a = a¹, then log_a (a) = 1

Also, since 1 = 2⁰ then log₂ (1) = 0

Similarly, log₃ (1) = 0 and log₁₀ (1) = 0, and as 1 = a⁰, then log_a (1) = 0.

See how you get on in this activity.

This activity and preceding section allows the following basic observations about logs to be made.

Before the days of the electronic calculator, logarithms were used everyday for multiplication and division and involved the use of log tables, as shown in Figure 1.

Figure 1 200 year old log tables

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A book showing 200 year old log tables

Figure 1 200 year old log tables

Nowadays, logarithms are mainly used in calculus or to find a linear function from an exponential one. You’ll be glad to hear that you don’t need a book of log tables now to work with logarithms.

On your calculator you should have a key called log (with the inverse function as 10^x or similar). This key allows you to take the log in base 10 of any positive number.

Try now in this quick activity.

In the next section, you’ll learn about three rules for operations involving logarithms.