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) About Me  (Show Source):
You can put this solution on YOUR website!

Let log%285%2C%282%29%29+=+x and log%285%2C%283%29%29+=+y

To figure out what log%2845%2C%28100%29%29 is in terms of x and y, we'll be using the Change of Base Rule

That rule is:
log%28b%2C%28x%29%29+=+%28log%28c%2C%28x%29%29%29%2F%28log%28c%2C%28b%29%29%29
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 c+%3C%3E+1
If c = 1 was the case, then we'd have a division by zero error.

Since we're dealing with log%2845%2C%28100%29%29, 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.
log%28b%2C%28x%29%29+=+%28log%28c%2C%28x%29%29%29%2F%28log%28c%2C%28b%29%29%29

log%2845%2C%28100%29%29+=+%28log%285%2C%28100%29%29%29%2F%28log%285%2C%2845%29%29%29

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.

log%2845%2C%28100%29%29+=+%282x%2B2%29%2F%282y%2B1%29 Apply the substitutions for x and y.

Answer by mccravyedwin(405) About Me  (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

log%2845%2C%28100%29%29%22%22=%22%22log%285%2C%28100%29%29%2Flog%285%2C%2845%29%29

Simplify the numerator:

log%285%2C%28100%29%29%22%22=%22%22log%285%2C%2810%5E2%29%29%22%22=%22%22log%285%2C%282%2A5%29%5E2%29%22%22=%22%222log%285%2C%282%2A5%29%29%22%22=%22%222%28log%285%2C%282%29%29%2Blog%285%2C%285%29%29%29%22%22=%22%222%28x%2B1%29%22%22=%22%222x%2B2

Simplify the denominator:

log%285%2C%2845%29%29%22%22=%22%22log%285%2C%285%2A9%29%29=log%285%2C%285%2A3%5E2%29%29%22%22=%22%22log%285%2C%285%29%29%2Blog%285%2C%283%5E2%29%29%22%22=%22%221%2B2log%285%2C%283%29%29%22%22=%22%221%2B2y

So, the answer is:

log%2845%2C%28100%29%29%22%22=%22%22%282x%2B2%29%2F%281%2B2y%29

Edwin