FOIL: 

These are our possible solutions. However, we need to test them.

:

The equation becomes . This is true, so  is a solution.

:

However, negative numbers do not have logarithms, so this equation is meaningless.  is not a solution, and  is the one and only solution. Since this is not one of our choices, the correct response is "The correct solution set is not included among the other choices."

The key to simplifying this problem is by using the Natural Logarithm Quotient Rule .

Using algebraic manipulation to bring each natural logarithm to one side, we obtain:

Using the logarithmic product rule , we simplify as follows:

Factoring this quadratic equation, we will obtain two roots.

The roots are  and . However, the domain of the logarithmic function is . That is to say, it is not defined for numbers less than or equal to 0. So our final answer is

Here we need to make use the power rule 

Recall that , so we have

We can rewrite  as , and then multiply each side by . Doing so, we get

This is just a quadratic equation with  replacing . Let us factor it just like a quadratic equation.

 in this case is a root with multiplicity of two, so there are two answers to this equality, both of them being .

In order to evaluate this equation, we have to do some algebraic manipulation first to get the exponential function isolated.

Here we employ the use of the logarithm base change formula .

We could convert either  or  to the other's base. Let's convert  to a logarithm with base 4.

Plugging this back in to the original equation,

First expand the logarithm using the product property:

We can evaluate the constant log on the left either by memorization, sight inspection, or deliberately re-writing 16 as a power of 4, which we will show here:

, so our expression becomes:

Now use the power property of logarithms:

Rewrite the equation accordingly.

Note that the 3rd terms becomes negative because the exponent is negative. We will use one last log property to finish simplifying:

Accordingly,

.

Now substitute and simplify:

First use the reversal of the logarithm power property to bring coefficients of the logs back inside the arguments:

Now apply this rule to every log in the formula and simplify:

Next, use a reversal of the change-of-base theorem to collapse the quotient:

Substituting, we get:

Now combine the two using the reversal of the logarithm product property:

Since we are looking for an inverse function, we start by swapping the x and y variables in our original equation.

Now we have to solve for y. To do this we have to work towards isolating y. First we remove the constant multiplier:

Next we eliminate the base on the right side by taking the natural log of both sides.

Subtract 1 and divide by 4: