We can combine the logs on the right side of the equation while also using the product rule on the log on the left side:

Since the logs have the same base, we can equate the algebraic equations equal to each other:

Now we can solve the quadratic equation:

Plugging this into the original equation doesn't produce errors, so it's our single solution.

First we take the natural log of both sides of the equation:

Remember, taking the natural log of  with an exponent equals that exponent, and the natural log of  is :

Form here, we solve algebraically:

First we start by subtracting  from each side:

Next, we rewrite the equation in exponent form:

Finally, we divide by :

First, we subtract  from each side:

Next, we divide each side by :

Now we rewrite the equation in exponent form:

And we finish using algebra:

First, we add  to each side:

Next, we take the exponent in the log and make it a coefficient:

And divide by the new coefficient:

Now we write the equation in exponent form:

The first thing we can do is combine all the log terms on the right side of the equation:

Next, we can take the coefficient from the left term and make it an exponent:

Now we can cancel the logs from both sides:

When we put  back into the original question, we don't have problems.  When we try it with  however, we get errors, so that's not a valid answer:

First, we take the coefficient, , and make it an exponent:

Now we can cancel the logs:

When we check our answers, however, we notice that  results in errors, so that's not a valid answer:

First, we combine the log terms on the left of the equation:

Now we can cancel the logs on each side:

We can subtract  from each side to set the equation equal to .  this will give us a nice quadratic equation to solve:

Notice that  is not a valid answer, because if we plug it into the original equation then we would be taking the log of a negative number, which we can't do.  Our only solution is:

The first thing we can do is move both log functions to one side of the equation:

Then we can combine the log functions (remember, when you add logs, we multiply the terms inside):

Now we can rewrite the equation in exponent form (and FOIL the multiplied terms):

We can collect all the terms on one side of the equation, and then solve the quadratic:

However, if we plug  into the initial equation, we would be taking the log of a negative number, which we can't do, so it's not a valid solution:

We first put both logs on one side of the equation:

Now we combine the log terms (remember, when we subtract logs we divide the terms inside):

We can now rewrite the equation in exponential form:

Anything raised to the  power is , and now we can solve algebraically: