18.3. Properties of Logarithms

18.3.1. Products

We will soon be solving exponential and logarithmic equations. For them we need to know some properties of logarithms. For example, how do we find the logarithm of the product of two numbers?

T1-83/84 screen for the exploration. You may have observed in the exploration that log 8 is the sum of log 4 and log 2. The calculator is set to show two decimal places.

▪ Exploration:

Try this. Find by calculator, using common logs:

log 4log 2log 8

  1. Can you find any relationship between the three numbers you have found?

  2. Can you express this relationship, if it turns out to be true, as a general rule?

  3. Repeat, using natural logs. Does the same relationship seem to hold?

We wish to write an expression for the log of the product of two positive numbers, say, M and N.

logb MN = ?

Let us substitute: M = bc and N = bd where b > 0 and b ≠ 1. Then

MN = bcbd = bc+d

by the laws of exponents. Writing this expression in logarithmic form, we have

logb MN = c + d

But, since bc = M, c = logb M. Similarly, d = logb N. Substituting, we have the following equation:

NOTE

The log of a product equals the sum of the logs of the factors.

Example 29:

Here are examples showing the log of a product.

  1. log 7x = log 7 + log x

  2. log 3 + log x + log y = log 3xy

NOTE

The log of ...

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