In order to condense the logarithmic expression, we use the following properties

As such

We expand logarithms using the same rules that we use to condense them.

Here we will use the quotient property

and the power property

.

Use the quotient property:

Rewrite the radical:

Now use the power property:

We start condensing our expression using the following property, which allows us to express the coefficients of two of our terms as exponents:

Our next step is to use the following property to combine our first three terms:

Finally, we can use the following property regarding subtraction of logarithms to obtain the condensed expression:

In order to solve this equation, we must apply several properties of logarithms. First we notice the term on the left side of the equation, which we can rewrite using the following property:

Where a is the coefficient of the logarithm and b is some arbitrary base. Next we look at the right side of the equation, which we can rewrite using the following property for the addition of logarithms:

Using both of these properties, we can rewrite the logarithmic equation as follows:

We have the same value for the base of the logarithm on each side, so the equation then simplifies to the following:

Which we can then factor to solve for :

Using the properties of logarithms, expand the logrithm one step at a time:

When expanding logarithms, division becomes subtration, multiplication becomes division, and exponents become coefficients.

.

Using the properties of logarithms,

the expression can be rewritten as

which simplifies to .

To expand

, use the quotient property of logs.

The quotient property states:

Substituting in our given information we get:

Exponentiate each side to cancel the natural log:

Square both sides:

Isolate x:

In order to expand the logarithmic expression, we use the following properties

As such

First, consolidate the left side into one logarithm:

convert to an exponent

subtract 64 from both sides

now we can solve using the quadratic formula: