SOLUTION: the question is: solve the following for x: 4log.sq(x) - log.2x = log.x^-2 sq(*) means square root i just didnt know the symbol for it i have no idea how to even start this o

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Question 875462: the question is: solve the following for x: 4log.sq(x) - log.2x = log.x^-2
sq(*) means square root i just didnt know the symbol for it
i have no idea how to even start this off.
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
it's easier to write this out than to type this out.
see the following picture:
in the following picture L stands for Log to the base of 10.
Unless specified, the base of 10 is always implied.
number 1 rewrites the problem.
number 2 uses log(a^b) = b*log(a) property of logs.
you get 4*log(sqrt(x)) = log(sqrt(x)^4)
number 3 uses log(a/b) = log(a) - log(b) property of logs.
you get log(sqrt(x)^4 - log(2x) = log(sqrt(x)^4/2x)
number 4 uses if log(a) = log(b), then a = b property of logs.
number 5 multiplies both sides of the equation by 2x.
number 6 replaces sqrt(x) with its equivalent form of x^(1/2)
number 7 uses (x^a)^b = x^(a*b) property of exponents.
you get (x^(1/2))^4 = x^(1/2 * 4) = x^2
number 8 uses x^a * x^b = x^(a+b) property of exponents.
it also uses the fact that x = x^1.
you get x^-2 * 2x = x^-2 * 2x^1 = x^-2 * x^1 * 2 = x^(-2+1) * 2 = x^-1 * 2 = 2 * x^-1 = 2x^-1
number 9 uses x^-a = 1/x^a property of exponents.
number 10 multiplies both sides of the equation by x.
you get x * x^2 = x^3 and 2/x * x = 2
number 11 takes the cube root of both sides of the equation.
your solution is x = cube root of 2 or its equivalent exponential form of x = 2^(1/3).
you can confirm the solution is correct by substituting cube root of 2 for x in the original equation.
i did and the solution is confirmed as correct.