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Differentiate Logs with Proof

In order to differentiate logs we must use the chain rule. The simplest type of log to differentiate is a natural log this can be done as shown below.

Differentiate Natural Logs
A natural log is a log to the base e.
d/dx (ln x) = 1/x

However if we want to differentiate ln(f(x)) we must use the chain rule to get

d/dx (ln(f(x)) = f'(x)/f(x)

Proof of Derivative of Natural Logs
Consider

y=ln(x)

then from the definition of a log we get

ey = x                   –(1)

Differentiate each side with respect to x (you need to use implicit differentiation for the left to get ey dy/dx) to get

ey dy/dx = 1

but from (1) we know that ey = x which we can substitute to get

x dy/dx =1

giving the derivative

dy/dx = 1/x

Log Laws: Adding and Subtracting Logs

It is often useful to combine several different logs, being added or subtracted, into a single log that can then be manipulated more easily.

For example it is easier to find log(6) than log(3)+log(4) – log(2)

In order to combine logs like this you need to use the following rules:
logc(a) + logc(b) = log(ab)
and
logc(a)-logc(b) = log(a/b)
Remember: all logs must be in the same base, you cant use this to add log2(a) + log3b

This rule can be proved quite simply as follows:

Let all logs be to the base c.

if log a = m and log b = n
then a=cm and b = cn
so ab = cmcn= cm+n
by taking logs we obtain
log(ab)=log(cm+n) = m + n
and by substituting in the values of m and n we get the required result
log(ab) = log a + log b

The result for log (a/b) can also be found in this way but I’ll leave that one for you to work out.

By David Woodford

Categories: maths Tags: , , , ,