5 min read
Mathew Wahome
Indices and Logarithms
Indices
An index (power) shows how many times a base is multiplied by itself: 2^4 = 2 x 2 x 2 x 2 = 16. In the Junior School CBC (KICD), Grade 9 Mathematics uses indices for large and small numbers and for growth problems.
Laws of indices
| Law | Rule | Example |
|---|---|---|
| Multiplication | a^m x a^n = a^(m+n) | 2^3 x 2^4 = 2^7 |
| Division | a^m / a^n = a^(m-n) | 5^6 / 5^2 = 5^4 |
| Power of a power | (a^m)^n = a^(mn) | (3^2)^3 = 3^6 |
| Zero index | a^0 = 1 | 7^0 = 1 |
| Negative index | a^-n = 1/a^n | 2^-3 = 1/8 |
Fractional index
a^(1/2) is the square root of a. 9^(1/2) = 3, and 8^(1/3) = 2 (cube root).
Solving equations
If 2^x = 32, write 32 = 2^5, so x = 5.
Logarithms
A logarithm is the inverse of an index. If a^x = N, then log_a N = x.
- 10^2 = 100, so log10 100 = 2.
- 2^5 = 32, so log2 32 = 5.
- log10 1 = 0, since 10^0 = 1.
Laws of logarithms
- log (AB) = log A + log B
- log (A/B) = log A - log B
- log (A^n) = n log A
Worked example
Simplify (2^5 x 2^3) / 2^6. = 2^(5+3-6) = 2^2 = 4.
Another: Find log10 20 + log10 5 = log10 100 = 2.
Try it
- Evaluate 4^3, 10^-2 and 25^(1/2).
- Solve 3^x = 81.
- Simplify (x^3 y^2)^2.
- Find log2 8 + log2 4.