Logarithms and Exponentials Mathematics Content

On the Reference sheet for this topic: Main Ideas: review of index laws what is a logarithm? relationship between y = ax and y = logax; change of base derivatives of logs and exponentials differentiation and integration of composite functions On the Reference sheet for this topic:

Index Laws Review

Cambridge

What is e? Like π, e is an approximation of an infinite decimal. It is between 2 and 3, (around 2.71828), and ex has the property that ex has a gradient of 1 at (0,1). It is called ‘Euler’s number’ after Leonard Euler, and has been found to 100 000 decimal places!

Transformations of Exponential graphs The red curve is y=ex. The blue curve is y=ex+3. Adding ‘3’ raises the y value by 3 (this is a ‘transformation’). Similarly, y=e(x-2) will move the curve 2 units to the right of y=ex.

Cambridge

Differentiating y = ex Similar results occur for other exponential functions. In general, d (ax) = kax where k is a constant dx

Groves

Function of a Function (Chain Rule)

Groves

Cambridge

Cambridge

Integrating Exponentials Consider integrating as the opposite process as differentiating. It follows that: and:

Groves

Cambridge

Logarithmic Functions Logarithms and exponentials are directly related. Using logarithms allows us to change the subject of exponential equations and solve them more easily. DEFINITION: If y=ax, then x is called the logarithm of y to the base a

Relationship between exponentials and logarithms The two graphs are reflections of each other, reflected across the line y=x, and their domains and ranges are exchanged.

We use two different type of logarithm - when you see ln notation it means logarithm using ‘logex’ while log is used for ‘log10x’.

Groves

Logarithm Laws

Other useful bits and pieces:

Given that log 5 3=0.68 and log 5 4=0.86, find:

Groves

Groves

Cambridge

Cambridge

Differentiating Logarithmic Functions

Function of a Function (Chain Rule)

Groves

Cambridge

Cambridge

Integration of Logarithmic Functions As integration is the inverse of differentiation,

Groves

Cambridge

You should be able to: integrate, differentiate and manipulate exponential and logarithmic functions complete a summary of the main points in this topic go back through other topics where we skipped questions involving this work complete HSC questions on this topic