SOLUTION: Solve the exponential or logarithmic equations. (Write your final answer in BOTH exact and approximate value). (a) 6^x-3 = 2^x (b) log(5x-6) = 2log x

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Solve the exponential or logarithmic equations. (Write your final answer in BOTH exact and approximate value). (a) 6^x-3 = 2^x (b) log(5x-6) = 2log x      Log On


   

Question 149398: Solve the exponential or logarithmic equations. (Write your final answer in BOTH exact and approximate value).
(a) 6^x-3 = 2^x
(b) log(5x-6) = 2log x
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
(a) 6^x-3 = 2^x
Take the log of both sides to get:
(x-3)log6 = xlog2
x(log6-log2) = 3log6
x(log3) = 3log6
x = 3log6/log3
x = 4.8927...
-------------------------
(b) log(5x-6) = 2log x
log(5x-6) = logx^2
x^2 = 5x-6
x^2-5x+6=0
(x-3)(x-2) = 0
x = 2 or x = 3
----------------------
Cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one to get you started.

a)

6%5E%28x-3%29+=+2%5Ex Start with the given equation.


log%2810%2C%286%5E%28x-3%29%29%29+=+log%2810%2C%282%5Ex%29%29 Take the log of both sides.


%28x-3%29log%2810%2C%286%29%29+=+x%2Alog%2810%2C%282%29%29 Rewrite both sides using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


x%2Alog%2810%2C%286%29%29-3log%2810%2C%286%29%29+=+x%2Alog%2810%2C%282%29%29 Distribute.


x%2Alog%2810%2C%286%29%29-3log%2810%2C%286%29%29-x%2Alog%2810%2C%282%29%29=0 Subtract x%2Alog%2810%2C%282%29%29 from both sides.


x%2Alog%2810%2C%286%29%29-x%2Alog%2810%2C%282%29%29=3log%2810%2C%286%29%29 Add 3log%2810%2C%286%29%29 to both sides.



x%28log%2810%2C%286%29%29-log%2810%2C%282%29%29%29=3log%2810%2C%286%29%29 Factor out the GCF "x"


x%28log%2810%2C%286%2F2%29%29%29=3log%2810%2C%286%29%29 Combine the logs using the identity log%28b%2C%28A%29%29-log%28b%2C%28B%29%29=log%28b%2C%28A%2FB%29%29


x%28log%2810%2C%283%29%29%29=3log%2810%2C%286%29%29 Divide.


x=3log%2810%2C%286%29%29%2F%28log%2810%2C%283%29%29%29 Divide both sides by log%2810%2C%283%29%29 to isolate x


x=log%2810%2C%286%5E3%29%29%2F%28log%2810%2C%283%29%29%29 Rewrite the expression using the identity y%2Alog%28b%2C%28x%29%29=log%28b%2C%28x%5Ey%29%29



x=log%2810%2C%28216%29%29%2F%28log%2810%2C%283%29%29%29 Raise 6 to the 3rd power to get 216


x=log%283%2C%28216%29%29 Use the change of base formula to rewrite the right side.


So the exact answer is x=log%283%2C%28216%29%29 which approximates to x=4.892789