SOLUTION: I need to solve the equation{{{4^(3x)=7^(x+1)}}}.My teacher uses handouts to teach,so I am stumped with no material to reference.Please help.Thanks.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: I need to solve the equation{{{4^(3x)=7^(x+1)}}}.My teacher uses handouts to teach,so I am stumped with no material to reference.Please help.Thanks.      Log On


   

Question 85515: I need to solve the equation4%5E%283x%29=7%5E%28x%2B1%29.My teacher uses handouts to teach,so I am stumped with no material to reference.Please help.Thanks.
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
4^(3x)=7^(x+1)
Since the variables are in the exponent, take the log of both sides to get:
(3x)log4 = (x+1)log7
(3log4)x = (log7)x + log7
(3log4-log7)x = log7
log[(4^3)/7]x = log7
0.961081934x = 0.84509804
x=0.8793194525...
===========
Cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
4%5E%283x%29=7%5E%28x%2B1%29 Start with the given equation

log%2810%2C%284%5E%283x%29%29%29=log%2810%2C%287%5E%28x%2B1%29%29%29 Take the log (which has a default base of 10) of both sides

3x%2Alog%2810%2C4%29=%28x%2B1%29log%2810%2C7%29 Rewrite the logarithms using the identity log%28b%2Cx%5Ea%29=a%2Alog%28b%2Cx%29

3x%2Alog%2810%2C4%29-%28x%2B1%29log%2810%2C7%29=0 Subtract %28x%2B1%29log%2810%2C7%29 from both sides


3x%2Alog%2810%2C4%29-%28x%2Alog%2810%2C7%29%2Blog%2810%2C7%29%29=0 Distribute log%2810%2C7%29

3x%2Alog%2810%2C4%29-x%2Alog%2810%2C7%29-log%2810%2C7%29=0 Distribute the negative

3x%2Alog%2810%2C4%29-x%2Alog%2810%2C7%29=log%2810%2C7%29 Add log%2810%2C7%29

x%283%2Alog%2810%2C4%29-log%2810%2C7%29%29=log%2810%2C7%29 Factor out an x

x%28log%2810%2C4%5E3%29-log%2810%2C7%29%29=log%2810%2C7%29 Rewrite the logarithm using the identity a%2Alog%28b%2Cx%29=log%28b%2Cx%5Ea%29

x%28log%2810%2C64%29-log%2810%2C7%29%29=log%2810%2C7%29 Raise 4 to the third power

x%28log%2810%2C%2864%2F7%29%29%29=log%2810%2C7%29 Combine the logs using log%28b%2Cx%29-log%28b%2Cy%29=log%28b%2C%28x%2Fy%29%29

x=log%2810%2C7%29%2F%28log%2810%2C%2864%2F7%29%29%29 Divide both sides by log%2810%2C%2864%2F7%29%29




x=0.845098%2F0.961082 Using log%2810%2C7%29=0.845098 and log%2810%2C%2864%2F7%29%29=0.961082 as approximations, evaluate the logs

x=0.879319 Divide


Check:

4%5E%283%2A0.879319%29=7%5E%280.879319%2B1%29 Plug in x=0.879319

4%5E%282.637957%29=7%5E%281.879319%29

38.7443=38.7443 works