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

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.