Exponentials and Logarithms

Exponentials

Exponential functions take the form $y=a^x$. Their features include:

• A $y$-intercept of $(0,1)$
• All $y>0$
• The $x$-axis as an asymptote.
• If $a>1$, then $y$ increases as $x$ increases, giving exponential growth.
• If $0<a<1$, then $y$ decreases as $x$ increases, giving exponential decay.

Modelling

Exponential functions are used to model situations where the rate of growth or decay is proportional to the current amount. This is because the derivative of $e^{kx}$ is $k{e}^{kx}$. A model with equation $y=A{e}^{kt}$ has:

• Initial value ($t=0$) of $y=A$
• A rate of change of $ky$.

Logarithms

Logarithms are the inverse of exponentials. Below are rules of logarithms:

• Converting: $a^b=c⟺{\log }_{a}c=b$
• Multiplication$:{\log }_b(xy)={\log }_{b}x+{\log }_{b}y$
• Division: ${\log }_b(\frac{x}{y})={\log }_{b}x-{\log }_{b}y$
• Exponentiation: ${\log }_b(x^y)=y{\log }_{b}x$

Other required knowledge:

• For any base, ${\log }_{a}a=1$ and ${\log }_{a}1=0$
• Logarithms in base $e$ are written as $\ln x$.
• Logarithms in base 10 are written as $\log x$.

A logarithmic function of the form $y={\log }_b(x)$ has:

• An $x$-intercept of $(1,0)$
• The $y$-axis as an asymptote.

Straightening graphs

Exponential

For a graph $y=A{b}^x$, taking $\ln$ on both sides gives:

$$\ln y=x\ln b+\ln A$$

Thus when plotting $\ln y$ against $x$, the gradient is $\ln b$ and $y$-intercept is $\ln A$.

Polynomial

For a graph $y=A{x}^b$, taking $\ln$ on both sides gives:

$$\ln y=b\ln x+\ln A$$

Thus when plotting $\ln y$ against $\ln x$, the gradient is $b$ and the $y$-intercept is $\ln A$.