Revision sheet

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 $x \div b$.

Asymptotes after shifts

• $y = a \cdot b^{x} + k$: horizontal asymptote $y = k$.
• $y = \log_b(x - h) + k$: vertical asymptote $x = h$, domain $x > h$.

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 $2n$.

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 $\log_b(M + N)$.

Change of base

$\log_a c = \dfrac{\ln c}{\ln a}$: the base goes below.

Check: $\log_2 5$ must be between 2 and 3, because $2^2 = 4$ and $2^3 = 8$.

Solving $e^{kx} = a$ (with $a > 0$)

1. Take $\ln$ of both sides: $kx = \ln a$.
2. Divide by $k$: $x = \dfrac{\ln a}{k}$.

Watch out: $e$ is not a factor. Never divide by $e$.

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.

Solving logarithmic equations

1. Combine into one logarithm (product, quotient, power rules).
2. Rewrite as a power, or compare arguments.
3. Solve, then check that every argument is positive.