Logarithm Introduction

From indices we have that 2³ = 8 where 2 is the base and 3 is the power. On the other hand, we can write that the log of 8 to base 2 is equal to 3, denoted thus

Log₂ 8 = 3

Also 5² = 25 which means that log₅25 = 2.

The log of any number N to base M is the index or power to which the base M must be raised to equal the number N.

i.e.    if x is the logarithm of a number N to base b then N = b^x

i.e.    if log_b N= x then N = b^x

e.g.    If       (i) log₂ 16 = 4, then 16 = 2⁴

(ii) log₅ 125 = 3, then 125 = 5³

(iii) log₉ 81 = 2, then 81 = 9²

(iv) log₂₅ 125 = 3/2 , then 125 = 25^3/2

(v) log₁₀ 1000 = 3, then 1000 = 10³

(iv) log₁₀ (1/10) = -1, then 1/10 = 10^-1

Conversely, if (i) 16 = 2⁴, then log₂ 16 = 4

(ii) 81 = 9², then log₉ 81 = 2

and  (iii) 125 = 25^3/2, then log₂₅ 125 = 3/2 and so on.

With the above expressions, we can say that logarithms and indices are inter-related.