Revision sheet

Exponential change: the same factor at every step

• Start $a$, factor $b$ per step: after $n$ steps, $a \cdot b^n$.
• Gain $p\%$: factor $1 + \frac{p}{100}$. Lose $p\%$: factor $1 - \frac{p}{100}$.

Watch out: adding the same amount each step is linear, not exponential.

Reading exponential functions

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

Watch out: negative exponent, positive value.

Rational exponents

• $b^{\frac{1}{n}} = \sqrt[n]{b}$; $b^{\frac{m}{n}} = \left(\sqrt[n]{b}\right)^m$.
• $b^{-r} = \dfrac{1}{b^{r}}$: the value stays positive.
• A factor $b$ per unit means a factor $b^{\frac12}$ per half-unit.

Writing an exponential function

• $f(x) = a \cdot b^x$: $a = f(0)$, $b = \dfrac{f(1)}{f(0)}$.
• Multiplied by $b$ every $n$ units: $f(x) = a \cdot b^{\frac{x}{n}}$.

Watch out: the starting value is $f(0)$, and it goes in front, not in the base.

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".

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 $e$.

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 $x = h + 1$.