# Exponentials and Logarithms

AS level content

• Know and use the function xa and its graph, where a is positive
• Know and use the function ex and its graph
• Know that the gradient of ekx is equal to kekx and hence understand why the exponential model is suitable in many applications
• Know and use the definition of logax as the inverse of ax, where a is positive and x≥0
• Know and use the function lnx and its graph
• Know and use lnx as the inverse function of ex
• Understand and use the laws of logarithms:

Logax + Logay = Loga(xy)

Logax - Logay = Loga(x/y)

kLogax  = Logaxk

• Solve equations of the form ax = b
• Use logarithmic graphs to estimate parameters in relationships of the form y=axn and y = kbx, given data for x and y
• Understand and use exponential growth and decay; use in modelling
• (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth);
• consideration of limitations and refinements of exponential models