Laws of indices and logarithms as their inverse
Exponents show repeated multiplication: aⁿ means n factors of a. The laws of indices are aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ. They extend naturally to other powers: a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^(1/n) is the nth root of a, so 8^(2/3) = 4. These rules are the key to simplifying expressions and to solving equations in which the unknown appears as a power.
A logarithm answers the question of what power the base must be raised to in order to give a number. If aˣ = N, then logₐ N = x, so log₂ 32 = 5 and log₁₀ 1000 = 3. The base must be positive and not equal to 1, and N must be positive. The log laws mirror the index laws: log(mn) = log m + log n, log(m/n) = log m − log n and log(mⁿ) = n log m. The change-of-base rule logₐ N = log N ÷ log a lets you work with whichever base is most convenient.
Why it matters
Logarithms turn multiplication into addition, which is why they were used for centuries to speed up long calculations. Today they appear in the pH scale in chemistry, the decibel scale for sound, earthquake magnitude scales and formulas for population growth and compound interest. In Class 12 calculus, the natural logarithm and the exponential function eˣ are among the most important functions you study.
Key terms
- Base
- The number that is raised to a power in an exponential or logarithmic expression.
- Logarithm
- The power to which a base must be raised to produce a given number.
- Negative exponent
- A power such as a⁻ⁿ, which means the reciprocal 1/aⁿ.
- Change of base
- A rule that rewrites a logarithm in one base as a ratio of logarithms in another.