]> Exercise 2.13

Exercise 2.13

Deduce them, that is, deduce: ln a b = ln a + ln b , and ( log a b ) * ( log b c ) = log a c .

Solution:

The natural logarithm function ln x is inverse to the exponential function. If in the first equation we take the exponential function of both sides we get exp ( ln a b ) on the left, which is a b , and exp ( ln a + ln b ) on the right.

Using the identity exp ( s + t ) = ( exp s ) * ( exp t ) the right hand side becomes a b as well.

The identity exp s t = ( exp s ) t implies the second identity in the case a = exp 1 by the following argument:
in that case log a b is ln b and log a c = ln c .
We can then apply the exponential function to both sides of the equation we want to prove and we get

exp ( ( ln b ) * ( log b c ) ) = exp ( ln c ) = c

By the identity the left hand side here becomes exp ( ln b ) ^ log b c , or b ^ log b c which is again c .

But this tells us that in general

log b c = ln c ln b

and the general claim, ( log a b ) * ( log b c ) = log a c follows immediately from this fact, by substitution.