How to express log52\log_5 2 in terms of a and b (Refer to qn)

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 ab1a 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(23)=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?

Source : Link , Question Author : MetaKnight35 , Answer Author : Zach Haney

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