Solve Logarithmic Equations - Pre-Calculus

To solve, simply remember that when you add logs, you multiply their insides.

Thus,

We must first use some properties of logs to rewrite the equation. First, using the power rule, which says

we can rewrie the left side of the equation, as below:

Now, we want to use the product property of logarithms to condense the right side into just one log, as below:

Because the logs are both base 10, we can simply set the insides equal, like this:

Now we have a polynomial to solve.

Using the quadratic formula to solve for x

By the property of the addition of logarithms with the same base

As such

To solve, simply remember that when add logs, you multiply their insides. Thus,

To solve, simply remember that when adding logs, you multiply their insides and when subtract logs, you divide your insides. You must use this in reverse to solve. Thus,

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:

Divide both sides by 25 to isolate the exponential function:

Take the natural log of both sides:

Solve for x:

The derifintion of logarithm is:

In this problem,

Therefore,

The natural logarithm and natural exponent are inverses of each other. Taking the of will simply result in the argument of the exponent.

That is

Now, , so

Using the properties of logarithms,

the expression can be rewritten as

which simplifies to .

Let's use the properties of logarithms to condense this equation. We will use the follwing three properties

Power Rule

Product Rule

Quotient Rule

Let's first use the power rule to rewrite the second term

Then, we'll use the product rule to combine the first threeterms

Lastly, we'll use the quotient rule to combine into one term

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

• 
• 

As such

Using the rules of logarithms,

Hence,

So exponentiate both sides with a base 10:

The exponent and the logarithm cancel out, leaving:

This answer does not match any of the answer choices, therefore the answer is 'None of the other choices'.

You need to know the Laws of Logarithms in order to solve this problem. The ones specifically used in this problem are the following:

Let's take this one variable at a time starting with expanding z:

Now y:

And finally expand x:

By the property of the sum of logarithms,

.

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 :

We start expanding our logarithm by using the following property:

Now we have two terms, and we can further expand the first term with the following property:

Now we only have two logarithms left with nonlinear terms, which we can expand using one final property:

Using this property on our two terms with exponents, we obtain the final expanded expression:

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:

The correct answer is

Properties of logarithms allow us to rewrite and as and , respectively. So we have

Again, we use the logarithm property

to get