SOLUTION: What is the answer to 3(7^2x) + 7^x+1 - 6 = 0?

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: What is the answer to 3(7^2x) + 7^x+1 - 6 = 0?      Log On


   

Question 773512: What is the answer to 3(7^2x) + 7^x+1 - 6 = 0?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I read what you wrote as 3%287%5E2x%29+%2B+7%5Ex%2B1+-+6+=+0 or maybe 3%2A7%5E%282x%29+%2B+7%5Ex%2B1+-+6+=+0
I am going to assume that you meant 3%2A7%5E%282x%29+%2B+7%5E%28x%2B1%29+-+6+=+0
I would change variables, defining y=7%5Ex.
y=7%5Ex means y%5E2=%287%5Ex%29%5E2=7%5E%282x%29
Also, 7%5E%28x%2B1%29=7%5Ex%2A7%5E1=7%2A7%5Ex
Substituting all those changes the equation becomes
3y%5E2%2B7y-6=0
Factoring we transform that equation into
%283y-2%29%28y%2B3%29=0
with solutions y=2%2F3 and y=-3.
You could get the same solutions using the quadratic formula, if you do not like factoring.
Going bacl to x as a variable, we see that
y=-3 would mean 7%5Ex=-3 and that is impossible,
so we are left with
y=2%2F3 which means 7%5Ex=2%2F3
Now we take logarithms of both sides.
If all we wanted is an exact solution, we could use base 7 logarithms:
log%287%2C7%5Ex%29=log%287%2C2%2F3%29 --> x=log%287%2C2%2F3%29 <--> x=log%287%2C2%29-log%287%2C3%29
However, if we wanted to get an approximate solution, we would use the more common base 10 logs.
Probably that is what the teacher expects. It would be
log%28%287%5Ex%29%29=log%28%282%2F3%29%29 --> x%2Alog%28%287%29%29=log%28%282%2F3%29%29 --> highlight%28x=log%28%28%222%2F3%22%29%29%2Flog%28%287%29%29%29 or highlight%28x=%28log%28%282%29%29-log%28%283%29%29%29%2Flog%28%287%29%29%29