SOLUTION: I keep messing up these, it's so confusing. Solve a)7^2x+1 = 4^x-2 b) 4^x = 3+18(4^-×) C) 9=logbase5 (x+100) +6

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: I keep messing up these, it's so confusing. Solve a)7^2x+1 = 4^x-2 b) 4^x = 3+18(4^-×) C) 9=logbase5 (x+100) +6      Log On


   

Question 1043601: I keep messing up these, it's so confusing.
Solve
a)7^2x+1 = 4^x-2
b) 4^x = 3+18(4^-×)
C) 9=logbase5 (x+100) +6
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Solve
a)7^2x+1 = 4^x-2
What are the exponents?
----------
b) 4^x = 3+18(4^-×)
Multiply thru by 4^x
4^(2x) = 3*4^x + 18
4^(2x) - 3*4^x - 18 = 0
--
Sub u for 4^x
4u^2 - 3u - 18 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-3x%2B-18+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-3%29%5E2-4%2A4%2A-18=297.

Discriminant d=297 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--3%2B-sqrt%28+297+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+297+%29%29%2F2%5C4+=+2.52921099245176
x%5B2%5D+=+%28-%28-3%29-sqrt%28+297+%29%29%2F2%5C4+=+-1.77921099245176

Quadratic expression 4x%5E2%2B-3x%2B-18 can be factored:
4x%5E2%2B-3x%2B-18+=+%28x-2.52921099245176%29%2A%28x--1.77921099245176%29
Again, the answer is: 2.52921099245176, -1.77921099245176. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-3%2Ax%2B-18+%29

===========
Ignore the negative solution.
4^x =~ 2.52921
x*log(4) = log(2.52921)
x = log(2.52921)/log(4)
x =~ 0.66934
==========================
C) 9=logbase5 (x+100) +6
3+=+log%285%2C%28x%2B100%29%29
log%285%2C5%5E3%29+=+log%285%2C%28x%2B100%29%29
x+100 = 125
x = 25