What is the value of x in the equation ln x + 6 ln 2x = 1?

To solve the equation ln x + 6 ln 2x = 1, we can begin by using the properties of logarithms.

First, recall the definition of logarithm: if y = ln a, then e^y = a. We will utilize this to rearrange our equation.

Start by rewriting ln 2x:

• ln 2x = ln 2 + ln x

This leads us to:

ln x + 6(ln 2 + ln x) = 1

Now distribute 6:

ln x + 6 ln 2 + 6 ln x = 1

This simplifies to:

7 ln x + 6 ln 2 = 1

Next, isolate the ln x term:

7 ln x = 1 – 6 ln 2

Now, divide both sides by 7:

ln x = rac{1 – 6 ln 2}{7}

To find x, we exponentiate both sides:

x = e^{rac{1 – 6 ln 2}{7}}

This gives us the final expression for x.