# Precalculus : Properties of Logarithms

## Example Questions

### Example Question #41 : Properties Of Logarithms

Solve the following logarithmic equation:

Explanation:

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

### Example Question #42 : Properties Of Logarithms

Solve this logarithm for

None of these.

Explanation:

Divide both sides by 25 to isolate the exponential function:

Take the natural log of both sides:

Solve for x:

### Example Question #43 : Properties Of Logarithms

Solve the following logarithmic equation.

Explanation:

In order to solve the logarithmic equation, we use the following property

As such

And converting from logarithmic form to exponential form

we get

Solving for x

And because the square of a difference is given as this equation through factoring

we have

which implies

### Example Question #44 : Properties Of Logarithms

Explanation:

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:

### Example Question #45 : Properties Of Logarithms

What is  equivalent to?

Explanation:

Using the properties of logarithms,

the expression can be rewritten as

which simplifies to .

### Example Question #11 : Solve Logarithmic Equations

Find the value of the sum of logarithms by condensing the expression.

Undefined

Explanation:

By the property of the sum of logarithms,

.

### Example Question #47 : Properties Of Logarithms

Condense the following logarithmic equation:

Explanation:

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:

### Example Question #41 : Properties Of Logarithms

What is another way of writing

?

Explanation:

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

Again, we use the logarithm property

to get

### Example Question #49 : Properties Of Logarithms

Write the expression in the most condensed form:

Explanation:

Use the Power property of Logarithms:

Rewrite the fractional exponent:

Condense into a fraction using the Quotient property of Logarithms:

Simplify: