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Please help an thank you so much /loga. (3x+5)-loga. (X-5)=loga 8

As I read the problem, what I see is log(3x+5) - log(x-5) = log(8)
where all the log's are base a.

Note that when the difference of logs is taken, it is the same as dividing the numbers.
This means the problem is log((3x+5)/(x-5)) = log(8), all log's in base A.

It is known that A^log(x) = x when the log is in base A.
This can be done to both sides, so we have (3x+5)/(x-5) = 8.

Multiplying both sides by (x-5) gives 3x+5 = 8x-40.
Adding -8x-5 to both sides gives -5x = = -45.
Dividing both sides by -5 gives x = 9.

Checking this gives log(32) - log(4) = log(8), all log's base a.
If we take the base a as 2, we get, since 32=2^5, 4=2^2, and 8=2^3, 5 - 2 = 3.
From this, the answer checks out.

If we don't worry about what a is, we can take convert log(32) - log(4) = log(32/4) = log(8).


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Scott A Wilson


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