Exponentials and Logarithms

Log Laws

$\log_a(xy) = \log_a x + \log_a y$ $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$ $\log_a(x^n) = n\log_a x$ $\log_a a = 1, \quad \log_a 1 = 0$

Change of Base

$\log_a x = \frac{\log_b x}{\log_b a}$

The Natural Exponential

$e \approx 2.71828...$. $\frac{d}{dx}(e^x) = e^x$.

$\ln x = \log_e x$. $e^{\ln x} = x$ and $\ln(e^x) = x$.

Solving Exponential Equations

$3^x = 20$ → $x\ln 3 = \ln 20$ → $x = \frac{\ln 20}{\ln 3} \approx 2.73$.

Solving Logarithmic Equations

$2\ln x - \ln(x+1) = \ln 3$ → $\ln\frac{x^2}{x+1} = \ln 3$ → $\frac{x^2}{x+1} = 3$ → $x^2 = 3x + 3$ → $x^2 - 3x - 3 = 0$.

$y = ab^t$ → $\ln y = \ln a + t\ln b$ (straight line with gradient $\ln b$).