SOLUTION: Let (((log(a,X))))=C and (((log(b,X))))=D. Find the general statement that expresses (((log(ab,X))))in term of C and D. I already know the answer is (CD)/(C+D) I cannot figur

Question 222790: Let (((log(a,X))))=C and (((log(b,X))))=D. Find the general statement that expresses (((log(ab,X))))in term of C and D.
I already know the answer is (CD)/(C+D)
I cannot figure out how to write a proof for this question.
I know all of the logarithmic principles as well but I can only come up with either (C)(D) as my answer or (C+D). I can't get a fraction.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
I found that it helps to work backwards for a bit to find out how to go forward. In this case, I broke down to get:

(used the change of base formula here)

(used the identity )

Then I realized that I needed to find substitutions for and (to somehow get 'x' in there). So I figured I'd solve for those respective expressions like so:

Start with the first equation.

Use the change of base formula.

Multiply both sides by .

Divide both sides by C.

So this means

---------------------

Start with the second equation.

Use the change of base formula.

Multiply both sides by

Divide both sides by D.

So this means

-------------------------------------------

Now let's get back to the main problem:

Start with the given expression.

Use the change of base formula.

Use the identity to break up the denominator.

Now here's where the substitutions come into play:

Plug in and

From here, it's just boils down to algebraically simplifying the expression:

Multiply the first inner fraction by .

Multiply the second inner fraction by .

Combine the lower fractions.

Multiply by the reciprocal of the lower fraction.

Multiply

Factor out the GCF from the denominator.

Cancel out the common terms.

Simplify

Rearrange the terms (trivial, but I figured that I'd match the book).

So this means where and

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