Exponential and Logarithmic Functions

Laws of indices

For $a, b > 0$:

• $a^m \times a^n = a^{m+n}$, $\quad a^m \div a^n = a^{m-n}$, $\quad (a^m)^n = a^{mn}$
• $a^0 = 1$, $\quad a^{-n} = \dfrac{1}{a^n}$, $\quad a^{\frac{1}{n}} = \sqrt[n]{a}$

Logarithms

$\log_a y = x$ means exactly the same as $a^x = y$. The common logarithm $\log y$ means $\log_{10} y$.

Law
$\log_a (MN) = \log_a M + \log_a N$	product
$\log_a \dfrac{M}{N} = \log_a M - \log_a N$	quotient
$\log_a M^n = n \log_a M$	power
$\log_a b = \dfrac{\log b}{\log a}$	change of base

Also $\log_a 1 = 0$ and $\log_a a = 1$.

Common mistake: $\log(M + N)$ is not $\log M + \log N$, and $\dfrac{\log M}{\log N}$ is not $\log M - \log N$.

Graphs

• $y = a^x$ ($a > 0$, $a \neq 1$) passes through $(0, 1)$ and always lies above the $x$-axis. It increases when $a > 1$ and decreases when $0 < a < 1$.
• $y = \log_a x$ passes through $(1, 0)$ and is defined only for $x > 0$.
• $y = a^x$ and $y = \log_a x$ are reflections of each other in the line $y = x$.

Solving exponential equations

Take logs of both sides:

$$3^x = 20 \Rightarrow x \log 3 = \log 20 \Rightarrow x = \frac{\log 20}{\log 3} \approx 2.73$$

Linear relations with logarithms

If $y = ka^x$, taking logs gives $\log y = \log k + x \log a$, so the graph of $\log y$ against $x$ is a straight line with slope $\log a$ and intercept $\log k$. DSE often gives such a straight-line graph and asks you to express $y$ in terms of $x$.

Real-life models

Growth or decay at a fixed rate $r$ per period: $A = P(1 \pm r)^t$. For example, a car losing 15% of its value every year is worth $P(0.85)^t$ after $t$ years.

Exam tip: when a question asks "after how many full years", solve the inequality, then round up to the next whole number and check it.