Home  18.013A  Chapter 2  Section 2.4 


Deduce them, that is, deduce: ln ab = 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 ab) on the left, which is ab, and exp(ln a + ln b) on the right.
Using the identity exp(s + t) = (exp s) * (exp t) the right hand side becomes ab as well.
The identity exp st = (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
and the general claim, (log_{a}b) * (log_{b}c) = log_{a}c follows immediately from this fact, by substitution.
