ย 

Logarithms for AQA A-level Maths

Logarithms

This page covers the following topics:

1. Basics of logarithms
2. The logarithmic function
3. Adding logarithms
4. Subtracting logarithms
5. Power of logarithms
6. Equations with logarithms

A logarithm is the opposite of a power. An exponential function such as aหฃ = b can be solved by rewriting it as logab = x, where a is the base of the logarithm. Where a is not given, it should be assumed to be 10. The change of base rule can be used to change the base of any logarithm.

Basics of logarithms

The natural logarithmic function is y = lnx, where lnx is the inverse of the natural exponential function, eหฃ. This can be used to solve equations with the two functions.

The logarithmic function

To add logarithms, the multiplication law must be used. The multiplication law states that logx + logy = logxy.

Adding logarithms

To subtract logarithms, the division law must be used. The division law states that logx โˆ’ logy = log(x/y).

Subtracting logarithms

The power rule of logarithms states that the logarithm of a power can be written as the exponent multiplied by the logarithm of the base.

Power of logarithms

All the rules of logarithms can be used to solve equations involving logarithms. If there is a single logarithm with the same base on both sides of the equation, then the logarithms can be omitted and the arguments can be set equal to each other.

Equations with logarithms

1

Solve for x: logโ‚‚(2xยฒ + 13x + 21) = logโ‚‚(x + 3) + 3.

Rearranging gives logโ‚‚(2xยฒ + 13x + 21) โˆ’ logโ‚‚(x + 3) = 3.
Using the division law, logโ‚‚((2xยฒ + 13x + 21)/(x + 3)) = 3.
By the definition of a logarithm, (2xยฒ + 13x + 21)/(x + 3) = 2ยณ.
Factorising the numerator gives (2x + 7)(x + 3)/(x + 3) = 6.
Cancelling out the common factor gives 2x + 7 = 6.
Then 2x = โˆ’1, therefore x = โˆ’1/2.

x = โˆ’1/2

Solve for x: logโ‚‚(2xยฒ + 13x + 21) = logโ‚‚(x + 3) + 3.

2

Find x for eหฃโปโน = 5, giving the answer in the form a + lnb, where a and b are integers.

Taking the natural logarithm of both sides gives x โˆ’ 9 = ln5 and x = 9 + ln5.

9 + ln5

Find x for eหฃโปโน = 5, giving the answer in the form a + lnb, where a and b are integers.

3

Simplify 4logโ‚ƒ(xโต).

Using the power rule of logarithms, 4logโ‚ƒ(xโต) = 5 ร— 4logโ‚ƒ(x) = 20logโ‚ƒ(x).

20logโ‚ƒ(x)

Simplify 4logโ‚ƒ(xโต).

4

Change the base of logโ‚ˆ5 to 2 and then evaluate it to 3 significant figures.

logโ‚ˆ5 = logโ‚‚5/logโ‚‚8 = 0.774 (to 3 significant figures)

0.774

Change the base of logโ‚ˆ5 to 2 and then evaluate it to 3 significant figures.

5

The graphs of y = 5หฃ and y = logโ‚…x are given on the same set of axes. Explain what can be deduced from this graph about the relationship of the two functions.

The two graphs are the reflections of the other in the line y = x. This means that it can be deduced from the graph that 5หฃ is the inverse of logโ‚…x.

One graph is an inverse of the other.

The graphs of y = 5หฃ and y = logโ‚…x are given on the same set of axes. Explain what can be deduced from this graph about the relationship of the two functions.

End of page

ย