Question 1009879
your equation is:


(x-8) / 3^8 = sqrt(3)


since sqrt(3) = 3^(1/2), your equation becomes:


(x-8) / 3^8 = 3^(1/2)


(x-8) = 3^y if and only if log3(x-8) = y


this means that (x-8) = 3^(log3(x-8))


your equation becomes:


3^(log3(x-8)) / 3^8 = 3^(1/2)


since 3^(log3(x-8)) / 3^8 is equivalent to 3^(log3(x-8) - 8), your equation becomes:


3^(log3(x-8)-8) = 3^(1/2)


this is true if and only if log3(x-8)-8) = 1/2


add 8 to both sides to get log3(x-8) = 8 + 1/2


simplify to get log3(x-8) = 17/2


this is true if and only if 3^(17/2) = x-8


add 8 to both sides of this equation to get x = 8 + 3^(17/2)


to confirm the solution is good, go back to the original equation and replace x with (8 + 3^(17/2)


the original equation is (x-8) / 3^8 = sqrt(3)


after you replace x, the equation becomes:


(8 + 3^(17/2) - 8) / 3^8 = 3^(1/2


the 8 - 8 cancels out and you are left with:


3^(17/2) / 3^8 = 3^(1/2)


since x^a / x^b = x^(a-b), this equation becomes:


3^((17/2)-8) = 3^(1/2)


make common denominators of the fractions to get:


3^((17/2)-(16/2)) = 3^(1/2)


combine like terms to get:


3^(1/2) = 3^(1/2)


this confirms the solution is correct.


the solution is x = 8 + 3^(17/2)


this confirms the solution is correct.


another way you could have solved it is as follows:


start with:

(x-8)/3^8 = 3^(1/2)


multiply both sides of this equation by 3^8 to get:


x-8 = 3^(1/2) * 3^8)


this becomes x-8 = 3^(17/2)


add 8 to both sides of this eqution to get:


x = 3^(17/2) + 8


this way is much quicker but the problem stated that everything had to be put in base 3 to an exponent form and so there was more to do in order to accomplish that.


either way you get the same answer which is always a good thing.