SOLUTION: If log₅2 = x and log₅3 = y, find log₄₅100 in terms of x and y.

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Question 1210375: If log₅2 = x and log₅3 = y,
find log₄₅100 in terms of x and y.
Found 2 solutions by math_tutor2020, mccravyedwin:
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

Let and

To figure out what is in terms of x and y, we'll be using the Change of Base Rule

That rule is:

where b is the original base and c is a new base to apply.
We can select any positive real number for c as long as
If c = 1 was the case, then we'd have a division by zero error.

Since we're dealing with , the original base is b = 45 and the input or argument to the log is x = 100.

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The variables x and y involve logs with base 5, so let's use c = 5.




Rewrite 100 as 2^2*5^2 and 45 as 3^2*5

Use the log rule log(A*B) = log(A)+log(B)

Applying log rule log(A^B) = B*log(A) to pull down the exponents.

When the log base and argument matches up, the result of the log is 1.

Apply the substitutions for x and y.
Answer by mccravyedwin(405)   (Show Source): You can put this solution on YOUR website!
If log₅2 = x and log₅3 = y,
find log₄₅100 in terms of x and y.
Get everything to log base 5



Simplify the numerator:



Simplify the denominator:



So, the answer is:



Edwin
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