SOLUTION: Hello, I need help solving this equation. I am supposed to show all steps to solve for x. log_(3)x+log_(3)(x+5)=9

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Hello, I need help solving this equation. I am supposed to show all steps to solve for x. log_(3)x+log_(3)(x+5)=9      Log On


   

Question 1206117: Hello, I need help solving this equation. I am supposed to show all steps to solve for x.
log_(3)x+log_(3)(x+5)=9
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
log(3,x)+log(3,(x+5))=9
log%283%2Cx%29%2Blog%283%2C%28x%2B5%29%29=9
log%283%2Cx%28x%2B5%29%29=9
if switch to exponential form
3%5E9=x%28x%2B5%29
You take it from here?

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

log%283%2C%28x%29%29%2Blog%283%2C%28x%2B5%29%29+=+9

log%283%2C%28x%28x%2B5%29%29%29+=+9 use log rule that log(A)+log(B) = log(A*B)

x%28x%2B5%29+=+3%5E9 convert from log form to exponent form

x%28x%2B5%29+=+19683

x%5E2%2B5x+=+19683

x%5E2%2B5x-19683+=+0

Apply the quadratic formula where a = 1, b = 5, c = -19683
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-5%2B-sqrt%28%285%29%5E2-4%281%29%28-19683%29%29%29%2F%282%281%29%29

x+=+%28-5%2B-sqrt%2825+%2B+78732%29%29%2F%282%29

x+=+%28-5%2B-sqrt%2878757%29%29%2F%282%29

x+=+%28-5%2Bsqrt%2878757%29%29%2F%282%29 or x+=+%28-5-sqrt%2878757%29%29%2F%282%29

x+=+137.818388 or x+=+-142.818388 both of which are approximate

Since the domain of log(x) is x > 0, it rules out x+=+%28-5-sqrt%2878757%29%29%2F%282%29+=+-142.818388 as a possible solution.

Therefore, we go with x+=+%28-5%2Bsqrt%2878757%29%29%2F%282%29+=+137.818388 as the only possible solution.

I used WolframAlpha to confirm the answer is correct.
https://www.wolframalpha.com/input?i=log_%283%29x%2Blog_%283%29%28x%2B5%29%3D9

Another way to confirm the answer is to let
f(x) = log(x)/log(3) + log(x+5)/log(3)
Note the use of the change of base formula.
Then you should find that f(137.818388) = 9 approximately.