SOLUTION: Let logX(base A) = C , and logX(baseB) = D. Find the general statement that expresses logX(base AB), in terms of C and D. Please help anyone. I think the answer is C+D or C+D/2.

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Question 254894: Let logX(base A) = C , and logX(baseB) = D. Find the general statement that expresses logX(base AB), in terms of C and D.
Please help anyone. I think the answer is C+D or C+D/2. All help is much appreciated.
Thanks
Answer by drk(1908) (Show Source):
You can put this solution on YOUR website!
Use several change of bases:
(i) log_A(x) = C
By change of base rules, we get
(ii) log(x)/log(A) = C
and solving for log(A) we get
(iii) log(A) = log(x)/C
--
(iv) log_B(x) = D
By change of base rules, we get
(v) log(x)/log(B) = D
and solving for log(B) we get
(vi) log(B) = log(x)/D
--
Now,
(vii) log_AB(x)
becomes
(viii) log(x)/log(AB)
which is expanded to
(ix) log(x) / (log(A) + log(B)).
So,
by substitution of (iii) and (vi) into (ix), we get
(x)
by adding the denominator fractions, we get
(xi)
factoring out the log(x) gives us
(xii)
and finally the answer as