SOLUTION: find the value of x if: log5 + log(x+2) - log(x-1) = 2

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Question 1062603: find the value of x if: log5 + log(x+2) - log(x-1) = 2
Found 2 solutions by Edwin McCravy, Theo:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%285%29%29+%2B+log%28%28x%2B2%29%29+-+log%28%28x-1%29%29%22%22=%22%222

Use the log property:   log%28%28A%29%29%2Blog%28%28B%29%29=log%28%28AB%29%29
on the first two terms on the left side:

log%28%285%28x%2B2%29%5E%22%22%29%29+-+log%28%28x-1%29%29%22%22=%22%222

Distribute 5%2A%28x%2B2%29to get 5x%2B10

log%28%285x%2B10%29%29+-+log%28%28x-1%29%29%22%22=%22%222

Use the log property:   log%28%28A%29%29-log%28%28B%29%29=log%28%28A%2FB%29%29
on the two terms on the left side:

log%28%28%285x%2B10%29%2F%28x-1%29%29%29%22%22=%22%222

Use the log property:  If log%28%28A%29%29=B then A=10%5EB

%285x%2B10%29%2F%28x-1%29%22%22=%22%2210%5E2

%285x%2B10%29%2F%28x-1%29%22%22=%22%22100

Multiply both sides by x-1

5x%2B10%22%22=%22%22100%28x-1%29

Distribute on the right:

5x%2B10%22%22=%22%22100x-100

Subtract 100x from both sides:

-95x%2B10%22%22=%22%22-100

Subtract 10 from both sides

-95x%22%22=%22%22-110

Divide both sides by -95

x%22%22=%22%22%28-110%29%2F%28-95%29

Reduce fraction and a negative divided 
by a negative is positive:

x%22%22=%22%2222%2F19

Edwin

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
start with:

log(5) + log(x+2) - log(x-1) = 2

subtract log(5) from both sides to get:

log(x+2) - log(x-1) = 2 - log(5)

since log(2) = 100, this becomes:

log(x+2) - log(x-1) = log(100) - log(5)

this is equivalent to:

log((x+2)/(x-1)) = log(100/5) which is equal to:

log((x+2)/(x-1)) = log(20)

this is true if and only if (x+2)/(x-1) = 20

solve for x to get x = 22/19.

replace x with 22/19 in your original equation and you will see that the equation is true.

you will get log(5) + log(60/19) - log(3/19) = 2

evaluate this using your calculator to get 2 = 2.

this confirms your solution is correct.

your solution is x = 22/19.