SOLUTION: log3 (2x – 1) – log3 (x – 4) = 2

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Question 278787: log3 (2x – 1) – log3 (x – 4) = 2
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your equations reads like this:

log%283%2C%282x-1%29%29+-+log%283%2C%28x-4%29%29+=+2

since log%28x%2Fy%29+=+log%28x%29+-+log%28y%29, your equation becomes:

log%283%2C%28%282x-1%29%2F%28x-4%29%29%29+=+2

The basic definition of logarithms states that y+=+log%28b%2Cx%29 if and only if b%5Ey+=+x.

by the basic definition of logarithms, log%283%2C%28%282x-1%29%2F%28x-4%29%29%29+=+2 if and only if 3%5E2+=+%282x-1%29%2F%28x-4%29

simplify to get:

9 = (2x-1)/(x-4)

multiply both sides of this equation by (x-4) to get:

9*(x-4) = (2x-1)

simplify to get:

9x - 36 = 2x - 1

subtract 2x from both sides of this equation and add 36 to both sides of this equation to get:

9x - 2x = -1 + 36

simplify to get:

7x = 35

divide both sides of this equation by 7 to get:

x = 5

your answer should be x = 5.

substitute in your original equation to see if this is true.

your original equation is:

log%283%2C%282x-1%29%29+-+log%283%2C%28x-4%29%29+=+2

substitute 5 for x to get:

log%283%2C%282%2A5-1%29%29+-+log%283%2C%285-4%29%29+=+2 which becomes:

log%283%2C9%29+-+log%283%2C1%29+=+2 which becomes:

log%283%2C%289%2F1%29%29+=+2 which becomes:

log%283%2C9%29 = 2 which is true if and only if 3%5E2+=+9 which it is.

your answer is:

x = 5