Changing Bases of Logarithms

February 21, 2019   

The topic of changing the base of a logarithm came up in class the other day and whilst it isn’t actually part of the Higher maths course, I thought it was worth a short post.

A numerical example

Let’s suppose that our calculator only has functions for log10(x) or ln(x) but we are asked to calculate log5(12). Well suppose we let y=log5(12)

then we have that 5y=12.
 Now we can take log10 of both sides

log10(5y)=log10(12)

and bring the power down ylog10(5)=log10(12)
 allowing us to finally rearrange for y: y=log10(12)log10(5).

Therefore we have log5(12)=log10(12)log10(5)

which we can calculate.

An algebraic example

The same process works in the general case too. Suppose y=logab

then ay=b
 and we can take logarithm with a different base logc(ay)=logc(b)
 move the power down ylogc(a)=logc(b),
 giving us y=logc(b)logc(a).

Putting all of this together gives us a general rule for changing the base of a logarithm.

loga(b)=logc(b)logc(a)