Solve the following equations for x. 1) 1/2 Inx= In (2x)?

Question

Solve the following equations for x. 1) 1/2 Inx= In (2x)?

Answer 1

Using log of a power theorem, move the 1/2 to become the exponent of x

1/2 ln x = ln (2x)

ln x ^ (1/2) = ln (2x)

since both have ln in front, get rid of ln on both sides

x ^ (1/2) = 2x

Square both sides

(x ^ (1/2)) ^ 2  = ( 2x ) ^2

x = 4x^2

0 = 4x^2 - x

Factor  0 = x ( 4x - 1)

so x = 0 or x = 1/4

Substitute back into the original to check

1/2 ln (0) = ln (2*0)

But you can't take ln of zero so the answer is x = 1/4

Good luck to you !!

Answer 2

First, recall that you can only take logarithms of positive numbers. Thus you need to solve for x in your equation *constrained to* x > 0



Next, ln(2x) = ln(x) + ln(2). Then you need to solve for x > 0 in

1/2 ln(x) = ln(x) + ln(2), or equivalently

1/2 ln(x) + ln(2) = 0

But 1/2 ln(x) = ln(sqrt(x)). Thus the equation becomes

ln(sqrt(x)) + ln(2) = 0.

Using again the fact that ln(sqrt(x)) + ln(2) = ln(2sqrt(x)), it follows that

ln(2sqrt(x)) = 0. You know that ln(y) = 0 if and only if y = 1, hence you need to solve for x > 0 such that

2sqrt(x) = 1, or x = 1/4. Since 1/4 > 0, the answer is

x = 1/4