Question 1009879: Solve the exponential equation by expressing each side as a power of the same base and then equating exponents.
(x-8)/(3^8) = sqrt ( 3)
The solution set is { }
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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.
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