SOLUTION: Solve the equation {{{log(16,(x))+log(8,(x))+log(4,(x))+log(2,(x))=6}}}

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Solve the equation {{{log(16,(x))+log(8,(x))+log(4,(x))+log(2,(x))=6}}}       Log On


   

Question 143992: Solve the equation log%2816%2C%28x%29%29%2Blog%288%2C%28x%29%29%2Blog%284%2C%28x%29%29%2Blog%282%2C%28x%29%29=6
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
log%2816%2C%28x%29%29%2Blog%288%2C%28x%29%29%2Blog%284%2C%28x%29%29%2Blog%282%2C%28x%29%29=6 Start with the given equation


Use the change of base formula to rewrite each individual log. Remeber the change of base formula is log%28b%2C%28x%29%29=log%2810%2C%28x%29%29%2Flog%2810%2C%28b%29%29


Rewrite 16 as 2%5E4. Rewrite 8 as 2%5E3. Rewrite 4 as 2%5E2


Rewrite the each denominator using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


Multiply both sides by the LCD 24%2Alog%2810%2C%282%29%29. Doing this will eliminate the denominators.


Distribute and multiply


log%2810%2C%28x%29%29%286%2B8%2B12%2B24%29=144%2Alog%2810%2C%282%29%29 Factor out the GCF log%2810%2C%28x%29%29


log%2810%2C%28x%29%29%2850%29=144%2Alog%2810%2C%282%29%29 Add


50%2Alog%2810%2C%28x%29%29=144%2Alog%2810%2C%282%29%29 Rearrange the terms


log%2810%2C%28x%5E50%29%29=log%2810%2C%282%5E144%29%29 Rewrite the expressions using the identity y%2Alog%28b%2C%28x%29%29=log%28b%2C%28x%5Ey%29%29


x%5E50=2%5E144 Since the logs have the same base, this means that the argument (the terms inside the log) are equal.


x=root%2850%2C2%5E144%29 Take the 50th root of both sides



So our answer is x=root%2850%2C2%5E144%29 which is approximately x=7.361501


note: x=root%2850%2C2%5E144%29 can be written as x=root%2850%2C2%5E144%29 =