# Logarithms for OCR A-level Maths

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

A logarithm is the opposite of a power. An exponential function such as aˣ = 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.

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

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

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

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.

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.

# 1

Solve for x: log₂(2x² + 13x + 21) = log₂(x + 3) + 3.

Rearranging gives log₂(2x² + 13x + 21) − log₂(x + 3) = 3.
Using the division law, log₂((2x² + 13x + 21)/(x + 3)) = 3.
By the definition of a logarithm, (2x² + 13x + 21)/(x + 3) = 2³.
Factorising the numerator gives (2x + 7)(x + 3)/(x + 3) = 6.
Cancelling out the common factor gives 2x + 7 = 6.
Then 2x = −1, therefore x = −1/2.

x = −1/2

# 2

Find x for eˣ⁻⁹ = 5, giving the answer in the form a + lnb, where a and b are integers.

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

9 + ln5

# 3

Simplify 4log₃(x⁵).

Using the power rule of logarithms, 4log₃(x⁵) = 5 × 4log₃(x) = 20log₃(x).

20log₃(x)

# 4

Change the base of log₈5 to 2 and then evaluate it to 3 significant figures.

log₈5 = log₂5/log₂8 = 0.774 (to 3 significant figures)

0.774

# 5

The graphs of y = 5ˣ and y = log₅x 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 5ˣ is the inverse of log₅x.

One graph is an inverse of the other.

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