AS/A-Level Mathematics - Logarithms

Logarithms are very important in solving problems related to growth and decay. Let's learn about it with examples in A-Level Maths!

Definition

If x and b are positive numbers and b ≠ 1 then the logarithm of x to the base b is the power to which b must be raised to equal x. It is written log_b x. In algebraic terms this means that

if y = log_b x then

x = b^y

The formula y = log_b x is said to be written in logarithmic form and x = by is said to be written in exponential form. In working with these problems it is most important to remember that y = log_b x and x = b^y are equivalent statements.

Example 1 : 

If log₄ x = 2 then

x = 42

x = 16

Example 2 : 

We have 25 = 5². 

Then log₅ 25 = 2.

Example 3 : 

If log₉ x = 1/2 then

x = 9 ^1/2

x = √9

x = 3

Example 4 : If log₂ (y/3) = 4 then

y/3 = 2⁴

y/3 = 16

y = 48

Solve the following:

(a) log₃ x = 4

(b) log_m 81 = 4

(c) log_x 1000 = 3

(d) log₂ (x/2) = 5

(e) log₃ y = 5

(f) log₂ 4x = 5

Ans:

(a) 81

(b) 3

(c) 10

(d) 64

(e) 243

(f) 8

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Drafted by Eunice (Maths)

Reference

https://en.wikipedia.org/wiki/Logarithm