Exponentials and logarithms

Exponential functions

An exponential function is \(f(x) = a \cdot b^x\) (\(b \gt 0\), \(b \ne 1\)). y-intercept \((0, a)\). The growth or decay factor \(b\): \(b \gt 1\) is growth, \(0 \lt b \lt 1\) is decay. Horizontal asymptote \(y = 0\). Domain \((-\infty, \infty)\); range \((0, \infty)\) when \(a \gt 0\).

Example: \(f(x) = 3 \cdot 2^x\): \(f(0) = 3\), \(f(2) = 12\), \(f(-1) = \dfrac{3}{2}\).

Compound and continuous interest

Compounded \(n\) times a year: \(A = P\left(1 + \dfrac{r}{n}\right)^{nt}\). Continuously: \(A = Pe^{rt}\), with the natural base \(e \approx 2.718\). Write \(r\) as a decimal.

Example: $1,000 at 6% for 5 years, monthly: \(1000(1.005)^{60} \approx 1348.85\) dollars. Continuously: \(1000e^{0.3} \approx 1349.86\) dollars (calculator).

Converting log and exponential form

\(\log_b x = y \iff b^y = x\). The base stays the base.

Example: \(\log_2 8 = 3 \iff 2^3 = 8\); \(\;5^{-2} = \dfrac{1}{25} \iff \log_5 \dfrac{1}{25} = -2\).

Exact logs to know

\(\log_b 1 = 0\), \(\log_b b = 1\), \(\log_b b^x = x\), \(b^{\log_b x} = x\). Common log: \(\log x = \log_{10} x\). Natural log: \(\ln x = \log_e x\).

Example: \(\log_2 8 = 3\); \(\;\log 1000 = 3\); \(\;\ln e^3 = 3\); \(\;\log_3 \dfrac{1}{9} = -2\).

Domain of a log

You can only take the log of a positive number. The graph of \(y = \log_b x\) passes through \((1, 0)\) and has a vertical asymptote \(x = 0\).

Example: \(f(x) = \log(x - 4)\): \(x - 4 \gt 0\), so the domain is \((4, \infty)\).

Watch out: \(\log 0\) and \(\log(-5)\) are undefined.

Log rules

For \(M, N \gt 0\) (and \(b \gt 0\), \(b \ne 1\)).

Example: \(\log_2(8x) = 3 + \log_2 x\); \(\;\log \dfrac{x}{100} = \log x - 2\); \(\;\ln x^5 = 5 \ln x\).

Watch out: \(\log(M + N) \ne \log M + \log N\), and \(\dfrac{\log M}{\log N} \ne \log M - \log N\).

Expanding and condensing

Expand: split products and quotients, bring exponents to the front. Condense: do it backwards (exponents go up first, then combine into one log).

Example: \(\log_3 \dfrac{x^2 y}{z} = 2\log_3 x + \log_3 y - \log_3 z\).

Example: \(2\ln x + \ln 3 - \ln y = \ln \dfrac{3x^2}{y}\).

Change of base

Change of base: \(\log_b x = \dfrac{\ln x}{\ln b} = \dfrac{\log x}{\log b}\). Use it to find any log on a calculator.

Example: \(\log_2 10 = \dfrac{\ln 10}{\ln 2} \approx 3.322\).

Solving exponential equations

Isolate the exponential part. If both sides can be written with the same base, set the exponents equal. If not, take \(\ln\) (or \(\log\)) of both sides and use the power rule.

Example: \(2^{x + 1} = 32 = 2^5\), so \(x = 4\). \(\;9^x = 27\): \(3^{2x} = 3^3\), so \(x = \dfrac{3}{2}\).

Example: \(5^x = 12\): \(x = \dfrac{\ln 12}{\ln 5} \approx 1.544\). \(\;3e^{2x} = 30\): \(e^{2x} = 10\), so \(x = \dfrac{\ln 10}{2} \approx 1.151\).

Doubling time and growth

With \(A = Pe^{rt}\), doubling means \(A = 2P\): \(2 = e^{rt}\), so \(t = \dfrac{\ln 2}{r}\).

Example: At 5% continuous growth: \(t = \dfrac{\ln 2}{0.05} \approx 13.86\) years.