Exponential and Logarithmic Functions

Notes

Arithmetic and geometric sequences

• Arithmetic: add the same common difference $d$ each step. $a_n = a_0 + dn$. These match linear functions.
• Geometric: multiply by the same common ratio $r$ each step. $g_n = g_0 r^n$. These match exponential functions.

Exponential functions

$f(x) = ab^x$ with $a \ne 0$, $b > 0$, $b \ne 1$.

• If $a > 0$: growth when $b > 1$, decay when $0 < b < 1$. The graph is always concave up and has a horizontal asymptote at $y = 0$.
• Over equal-length input intervals, outputs change by equal factors (the ratio is constant). That's how you spot exponential data.
• Percent change: growth by $r\%$ per period gives $\displaystyle b = 1 + \frac{r}{100}$; decay gives $\displaystyle b = 1 - \frac{r}{100}$.
• Equivalent forms: $f(x) = ab^x$ can be rewritten, for example $b^{x+k} = b^k b^x$ (a horizontal shift is the same as a vertical stretch). Continuous growth uses $e$: $f(t) = ae^{kt}$.

Logarithms

$\log_b c = a$ means $b^a = c$. A log answers "what power?"

• $\log_b x$ is the inverse of $b^x$: domain $x > 0$, vertical asymptote at $x = 0$.
• Rules: $\log_b(xy) = \log_b x + \log_b y$, $\displaystyle \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$, $\log_b(x^n) = n\log_b x$.
• Change of base: $\displaystyle \log_b x = \frac{\ln x}{\ln b}$.
• Over equal-length output intervals of a log function, inputs change by equal factors. Logs grow, but slower and slower (concave down when $b > 1$).

Solving equations

• Exponential equation: take a log of both sides. $5 \cdot 2^x = 40 \Rightarrow 2^x = 8 \Rightarrow x = 3$.
• Log equation: rewrite in exponential form, then check that every log in the original has a positive input.

Semi-log plots and modeling

On a semi-log plot (the $y$-axis is logarithmic), exponential data appears as a line. If $y = ab^x$, then $\log y = \log a + x\log b$.