## Logarithms Properties

#### Property 1

loga1 = 0

Recall that if a ≠0, a0 = 1 therefore, loga1 = 0

Hence, the logarithm of 1 to any base is 0.

Example

log21 = 0; because in the exponential equation

we know that 20 = 1

log1/21 = 0; because in the exponential equation

we know that (½)0 = 1

The logarithm of unit to any non–zero base is zero ( where unity = 1)

#### Property 2

logaa = 1

Recall that in exponents a1 = a ⇒logaa =1

(where a ≠0, a > 0)

Example

log1010 = 1 because in the exponential equation

we know that 101 = 10

log77 = 1 because in the exponential equation

we know that 71 = 7

logae = 1 because in the exponential equation

we know that e1= e

The logarithm of any non–zero positive number to the same base is unity.

#### Property 3

logaax = x

Recall that ax = ax

Example

Since you know that 34 = 34, you can write the logarithm equation as log334 = 4

#### Try these questions

Find the following

1. log381
Answer: log381 = log334 = 4

2. log61
6x = 1   ⇒ x = 0

3. log1x = 23 Find x
Answer: log1x = 23   ⇒ 123 = 1
∴ x = 1

4. log2256
Answer: log2256 = log228 = 8

5. log525
Answer: log525 = log552 = 2

6. If logx8 = 3 then find x
Answer: logx8 = 3   ⇒ x3 = 8
x3 = 23
x = 2