In my textbook, I came across this interesting question which I am currently struggling to solve:
If log62=a and log53=b, express log52 in terms of a and b
The solution given is ab1−a but I do not know the working behind this. What is it?
An alternative, although I agree with the exponential approach, too!
We’re given: log62=a and log53=b
We want: log52
We must recall our logarithms rules. There are too many bases happening here, so let’s fix that!
The change of base formula gives us log62=log52log56 — I thought to do this because we’re looking for all base 5. Interesting! It has what we’re looking for, i.e. log52 !
Another rule from logarithms tells us: log56=log5(2∗3)=log52+log53. Aha! We see again our lovely longed for log52!!
Reviewing what we have: a=log62=log52log56 and log56=log52+log53 (=log52+b)
So, we have from the first part alog56=log52 — we have a substitution from above we can do!
alog56=log52 => a(log52+b)=log52. Distributing appeals to you. And factoring somewhere down the road.
Do you see how to arrive at the final solution?