Use and Properties of Indices and Logarithms

Understanding Indices

Indices, also known as powers or exponents, are a way of expressing how many times a number (the base) is multiplied by itself.
The expression a^n is read as “a to the power of n” where a is the base and n is the exponent.
Any non-zero number raised to the power of zero is 1, i.e. a^0 = 1.
To multiply powers with the same base, add the exponents: a^m * a^n = a^(m+n).
To divide powers with the same base, subtract the exponents: a^m / a^n = a^(m-n).
Raising a power to another power means to multiply the exponents: (a^m)^n = a^(mn).
A negative exponent means taking the reciprocal of the base to the positive power, i.e. a^(-n) = 1/a^n.

Logarithms are the inverse of exponential relationships; they rewrite multiplications as additions and divisions as subtractions.
The logarithm of a number x to a given base b, represented as log_b x, answers the question: “what exponent do we need to raise b to, to get x?”.
Basic properties of logs include log_b 1 = 0 and log_b b = 1.
The sum of logarithms equals the logarithm of their product, known as the Product Law: log_b (mn) = log_b m + log_b n.
The difference of logarithms equals the logarithm of their quotient, known as the Quotient Law: log_b (m/n) = log_b m - log_b n.
The exponent inside a logarithm can be relocated as a multiplier outside, known as the Power Law: log_b m^n = n log_b m.

The change of base formula helps compute logarithms in inconvenient bases. It is given by log_b a = log_c a / log_c b, where c can be any positive number different from 1.
Natural logs (base e) and common logs (base 10) are particularly useful; calculators often have buttons for these bases.
The change of base formula can simplify calculations and make logarithms easier to compare and compute.

Indices and logarithms are used in many scientific, mathematical, and real-world contexts.
Exponential growth and decay, population dynamics, radioactive decay, the Richter scale for earthquake severity, the pH scale for acidity, and sound intensity are all modelled using exponential and logarithmic functions.
Understanding and manipulating these functions provide essential tools for solving complex problems in various fields.

An excellent pathway to mastery in this topic is consistent practice and problem-solving across a range of difficulty levels.