SOLUTION: 5^x=3-5^(2x) Please help! Sorry I've been asking a lot of questions but I have a test tomorrow and we had a supply teacher who didn't know the material. The teacher left handout

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: 5^x=3-5^(2x) Please help! Sorry I've been asking a lot of questions but I have a test tomorrow and we had a supply teacher who didn't know the material. The teacher left handout      Log On


   

Question 1098778: 5^x=3-5^(2x)
Please help! Sorry I've been asking a lot of questions but I have a test tomorrow and we had a supply teacher who didn't know the material. The teacher left handouts without a lesson so I've been figuring things out on my own. More likely I'm really tired and can't concentrate. Sorry for the inconvenience.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!

With the "5^2x" and "5^x" terms in the equation, this can be viewed as a "quadratic" equation in which the "variable" is 5^x, because %285%5Ex%29%5E2+=+5%5E%282x%29.

So rewrite the equation as
5%5E%282x%29+%2B+5%5Ex+-+3+=+0

If it helps ease the confusion, you can define a new variable u+=+5%5Ex and write the equation as u%5E2+%2B+u+-+3+=+0

The quadratic does not factor, so use the quadratic formula:
5%5Ex+=+%28-1%2B-sqrt%2813%29%29%2F2

Since 5^x is never negative, we choose the positive solution: 5%5Ex+=+%28-1%2Bsqrt%2813%29%29%2F2

Then to solve for x, since it is an exponent, we take logs of both sides:
x%2Alog%285%29+=+log%28%28-1%2Bsqrt%2813%29%29%2F2%29
x+=+log%28%28-1%2Bsqrt%2813%29%29%2F2%29%2Flog%285%29

Answer by ikleyn(52887) About Me  (Show Source):
You can put this solution on YOUR website!
.
For many solved problems of this type (your samples) see the lesson
    - Solving exponential equations
in this site.


Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic "Exponential equations".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.