3. The Logarithm Laws

by M. Bourne

Since a logarithm is simply an exponent which is just being written down on the line, we expect the logarithm laws to work the same as the rules for exponents, and luckily, they do.

ExponentsLogarithms
`b^m × b^n = b^(m+n)` ` log_b xy = log_b x + log_b y`
`b^m ÷ b^n = b^(m-n)` `log_b (x/y) = log_b x − log_b y`
`(b^m)^n = b^(mn)` `log_b (x^n) = n log_b x`
`b^1 = b` `log_b (b) = 1`
`b^0 = 1` `log_b (1) = 0`

Note: On our calculators, "log" (without any base) is taken to mean "log base 10". So, for example "log 7" means "log107".

Examples

1. Expand

log 7x

as the sum of 2 logarithms.

2. Using your calculator, show that

`log (20/5) = log\ 20 − log\ 5`.

3. Express as a multiple of logarithms: log x5.

Note 1: Each of the following is equal to 1:

log6 6 = log10 10 = logx x = loga a = 1

The equivalent statements, using ordinary exponents, are as follows:

61 = 6

101 = 10

x1 = x

a1 = a


Note 2: All of the following are equivalent to `0`:

log7 1 = log10 1 = loge1 = logx 1 = 0

The equivalent statments in exponential form are:

70 = 1

100 = 1

e0 = 1

x0 = 1

Exercises

1. Express as a sum, difference, or multiple of logarithms:

`log_3((root(3)y)/8)`

2. Express

2 loge 2 + 3 loge n

as the logarithm of a single quantity.

Note: The logarithm to base e is a very important logarithm. You will meet it first in Natural Logs (Base e) and will see it throughout the calculus chapters later.


3. Determine the exact value of:

`log_3root(4)27`

4. Solve for y in terms of x:

log2x + log2y = 1

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