Revise Exponents and logarithms for IB Mathematics: Analysis and Approaches (1.2)

Before logarithms we must be completely confident with the laws of exponents. These rules govern every manipulation of powers, and crucially they hold for rational exponents (fractions) exactly as they do for whole numbers. A negative index signals a reciprocal; a fractional index signals a root. Reading these two facts in both directions is what makes non-calculator simplification fast.

For a base $a$ and exponents $m, n \in \mathbb{Q}$:

• Product rule: $a^m \times a^n = a^{m+n}$
• Quotient rule: $a^m \div a^n = a^{m-n}$
• Power rule: $(a^m)^n = a^{mn}$
• Zero index: $a^0 = 1$ (for $a \neq 0$)
• Negative index: $a^{-n} = \dfrac{1}{a^n}$
• Rational index: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

A logarithm is the inverse of an exponential. The statement $\log_a x = y$ asks: 'to what power $y$ must I raise the base $a$ to obtain $x$?' That is exactly the exponential statement $a^y = x$. Learning to flip instantly between these two forms is the single most useful skill in this topic.

$\log_a x = y \iff a^y = x$

Valid for base $a > 0,\ a \neq 1$ and argument $x > 0$.

Two values worth knowing on sight:

• $\log_a a = 1$ \quad (since $a^1 = a$)
• $\log_a 1 = 0$ \quad (since $a^0 = 1$)

Because logs invert powers, each exponent law has a matching logarithm law. These let you combine several logarithms into one, or expand one into several — the step that usually unlocks a logarithmic equation. They are in the formula booklet, but on Paper 1 you must apply them without hesitation.

For base $a$ and positive arguments $x, y$:

• Product law: $\log_a(xy) = \log_a x + \log_a y$
• Quotient law: $\log_a\!\left(\dfrac{x}{y}\right) = \log_a x - \log_a y$
• Power law: $\log_a(x^k) = k\log_a x$

Two logarithms appear so often they get their own notation: the common logarithm $\log x = \log_{10} x$, and the natural logarithm $\ln x = \log_e x$ with $e \approx 2.718$. When an equation mixes logarithms of different bases, the change-of-base formula rewrites them all in one convenient base so ordinary algebra can take over.

$\log_b a = \frac{\log_c a}{\log_c b}$

for any valid new base $c$ (often $c = 10$ or $c = e$). A useful special case: $\log_b a = \frac{1}{\log_a b}.$

On Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and follow-through (FT) means a slip early on need not cost you the marks that depend on it, provided your later working is correct on your own figures. But that protection only exists if the method is written down. Study how each mark below is earned by one specific line when we solve an exponential equation with the unknown in both exponents.

Exponents and logarithms underpin far more of the course than this one topic. The exponential and logarithmic functions $y = a^x$ and $y = \log_a x$ build directly on the inverse relationship you have just met, and the number $e$ and the natural log reappear throughout differentiation and integration. Compound interest, growth and decay models, and the solving of any equation with the unknown in an exponent all reduce to the same routine: use the laws to isolate the unknown, switch between exponential and logarithmic form, and check the domain. Master that routine now and the functions and calculus that follow become variations on a method you already own.