Understand and use the laws of logarithms: log_a x + log_a y = log_a(xy); log_a x – log_a y = log_a(x/y); k log_a x = log_a(xᵏ) (including, for example, k = –1 and k = –½)

Logarithms are the inverse of exponentials, and their laws are essential for simplifying expressions and solving equations involving logs. The three core laws—product, quotient, and power—allow you to combine or break apart logarithmic terms. For example, log_a x + log_a y = log_a(xy) turns addition into multiplication inside the log, which is crucial when solving equations where the variable appears inside multiple logs. These laws are not just abstract rules; they are used extensively in calculus (differentiation of log functions), exponential modelling, and even in topics like radioactive decay or compound interest.

In the Edexcel A-Level specification, you are expected to apply these laws fluently, including cases where the multiplier k is negative or fractional. For instance, -log_a x = log_a(1/x) and -½ log_a x = log_a(1/√x). Understanding these forms is vital for solving equations like log_a x - log_a y = log_a(x/y) and for simplifying expressions before differentiation or integration. Mastery of these laws also underpins the change of base formula and natural logarithms, which appear later in the course.

This topic is a gateway to more advanced work: without a solid grasp of log laws, you will struggle with exponential equations, logarithmic differentiation, and even mechanics problems involving exponential decay. The key is to practice rewriting expressions in different forms—for example, converting a sum of logs into a single log—so that you can solve equations that would otherwise be unsolvable by standard algebraic methods.