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Logarithms and Exponentials

Logarithms and Exponentials involve studying the following:

Know and use a to the power of x and e to the power of x and their graphs;

Know that the gradient of e^(kx) is equal to ke^(kx) and hence understand why the exponential modal is suitable in many applications;

Know and use the definitions of log(a)x and its inverse, a^x where a is positive and x>=0;

Know and use the function natural log x and its graph.

Know and use x in the inverse of ln x;

Understand and use the Laws of Logarithms;

Solve equations of the form a^x = b;

Use logarithmic graphs to estimate parameters in relationships of the form y=ax^n and y = kb^x given data for x and y;

Understand and use exponential growth and decay and use these in modelling.

Logarithms Video 1

Powers and Indices

Graphs of Exponents and Logs

Watch Logarithms Video 1 and then have a go at this exercise.

Logarithms Video 2

Exponential Function

What is e?

A very brief history on the value of e.

Logarithms Video 3

Index and logarithm forms

Logarithms and Other Tables

A copy of a historic document: Logarithms and Other Tables for Schools by F. Castle. A touch of nostalgia for those of us of a certain age. The back few pages act as a reminder of how to use the tables.

Logarithmic Laws Video

Using the Laws of Logarithms to combine expressions together to present a simplified version.

Changing Logarithms from One Base to Another

Changing logarithms from one base to another can be done on a calculator where you can just put the base number in as required. However, for those of us possessing older calculators (or indeed, log tables booklets), this method will come in handy.

Writing Indices with Mixed Bases as a Single Index

Writing indices with mixed bases as a single index is simple when you understand that logarithms are the opposite to powers. So in the same way as we can undo addition using subtraction, multiploication using division, we can undo powers using logarithms and vice versa.

Finding the value of x

Log base 10 problem involving moving to the power of 10

Find the value of x by raising both sides to the power of ten and then cancelling down.

Natural Logarithm Problems

Solve a natural logarithm problem by raising to the value of e and then rearranging the equations to find the exact value of the unknown.

Log Problems with Quadratics

Using natural logarithms, substitution and quadratic equations to find the unknown value.

Using quadratic equations and substitution to find the two values of x in the solution of this problem.