Revision sheet

Reading exponential functions

$a$: value at $x = 0$; $b$: factor per step; $b^{-1} = 1/b$.

Watch out: negative exponent, positive value.

Logarithms are exponents

• $\log_b x = y$ means $b^y = x$.
• $\log x$ means $\log_{10} x$.
• The solution of $b^x = c$ is $x = \log_b c$.

Watch out: $\log_b x$ is never $x \div b$.

The number $e$

• $e \approx 2.718$, the limit of $\left(1 + \frac{1}{x}\right)^x$.
• Continuous growth at rate $r$: $A(t) = A_0 \, e^{rt}$.
• In $a \, e^{kt}$, the factor per unit of $t$ is $e^{k}$.

Watch out: "continuously at $r$" is not "by $1 + r$ once a year".

Logarithmic functions

• $y = \log_b x$ is the inverse of $y = b^x$.
• Through $(1, 0)$ and $(b, 1)$; domain $x > 0$; asymptote $x = 0$.
• $y = \log_b(x - h)$: asymptote $x = h$; crosses the $x$-axis at $x = h + 1$.

Properties of logarithms ($M, N > 0$)

• Product: $\log_b(MN) = \log_b M + \log_b N$.
• Quotient: $\log_b \frac{M}{N} = \log_b M - \log_b N$.
• Power: $\log_b(M^p) = p \log_b M$.

Watch out: there is no rule for $\log_b(M + N)$.

Solving $a \cdot b^x = c$

1. Isolate the power: $b^x = \dfrac{c}{a}$.
2. Take $\ln$: $x \ln b = \ln \dfrac{c}{a}$.
3. Divide: $x = \dfrac{\ln(c/a)}{\ln b}$.

Watch out: never multiply $a$ into the base.

Half-life and doubling time

• Half-life $h$: $A_0 \left(\tfrac{1}{2}\right)^{\frac{t}{h}}$. Doubling time $d$: $P_0 \cdot 2^{\frac{t}{d}}$.
• $\frac{t}{h}$ = number of half-lives.

Watch out: $n$ half-lives divide by $2^n$, not by $2n$.