- A-Level Maths

# Exponentials and Logarithms

An **exponential function** is one a constant is raised to the power of a variable:

The larger the coefficient, the steeper the graph

All exponential functions in the form y = a^x pass through (0, 1)

The value of the function decreases as x tends to 0

Functions in the form y = a^x where 0 < a < 1 are the other way around:

The gradient graphs of an exponential function are always very similar, but when **a = 2.72 ( e), the gradient graph is exactly the same:**

For all real values of x:

if f(x) =e^x, f'(x) = e^x

if f(x) =e^kx, f'(x) = ke^kx

Functions in *e* can be used to model growth or decline where the rate of increase in number is proportional to the number.

## Logarithms

**Logarithms are the inverse of exponential functions. **

Just as indices laws apply to exponentials, there are a series of laws that apply to logarithms - these are known as the **log rules **or **laws of logarithms:**

When working with difficult equations with exponents, it can be helpful to apply logarithms to both sides:

### The Natural Logarithm, ln

The natural logarithm, ln, is the logarithm in base *e*. The graph of ln(x) is reflection of e^x in the line y=x:

The y=axis is an asymptote in this graph, meaning that

ln(x) is only defined when x > 0

### Logarithms and Non-Linear Data

Often, it is helpful to model non-linear data as linear. This can be achieved by taking logs of one or both sides of the equation:

If the equation is in the form

**y = ax^n**, a graph of**log y against log x**will give a straight line where log a is the y intercept and n the gradient.

If the equation is in the form

**y = ab^x**, a graph of l**og y against x**will give a straight line where log a is the y intercept and log b the gradient.