Question 998682
<br>
solve for x
x= 6/8 (log(base6) sqrt(108) - 1 log(base6) (2/sqrt(3))+ 2 log (base6) 9 + log(base6) 64)
I am not sure how to find X, I would appreciate if someone can help me. I will provide my step.
x=6/8 (log(base6) sqrt(108) - 1 log(base6) (2/sqrt(3))+ 2 log (base6) 9 + log(base6) 64)
=6/8 (log(base6) sqrt(108)/ (2/sqrt3) + log(base 6) 9^2 + log(base6) 64)
=6/8 (log(base6) sqrt(108) / (2/sqrt3) + log(base 6) (81)(64)
=6/8 (log(base6) 9 + log(base6) 5184)

=6/8 (log(base6) (9)(5184)
=6/8 (log(base6) 46656)
= log(base6) 34992<br>
------------------------------------------------------------------<br>
Your work is fine up to the last step.<br>
It will be easier to find the answer if you leave the expression in factored form instead of multiplying all the numbers together.<br>
The entire expression is of the form<br>
(6/8) (... extended expression in a form equivalent to "log (base6) of A")<br>
Let's drop all the "log (base 6)" notations and just look at the "A".  Using basic log rules, A is as follows:<br>
{{{((sqrt(108))(9^2)(64))/(2/sqrt(3))}}}<br>
Performing basic arithmetic WITHOUT multiplying all the numbers together....<br>
{{{(6sqrt(3))(9^2)(64)(sqrt(3)/2)}}}<br>
{{{(6)(3)(9^2)(32)}}}<br>
And now remembering that we are looking for the value of the log base 6 of this number....<br>
{{{(6)(3)(3^4)(2^5)=(6)(3^5)(2^5)=(6)(6^5)=6^6}}}<br>
Finally, then....<br>
{{{x=(6/8)log(6,(6^6))=(3/4)(6)=9/2}}}<br>
ANSWER: 9/2<br>