Precalculus : Properties of Logarithms

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #41 : Properties Of Logarithms

Solve the following logarithmic equation:

Possible Answers:

Correct answer:

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

Possible Answers:

None of these.

Correct answer:

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.  

 

 

Possible Answers:

Correct answer:

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

Express the log in its expanded form:

Possible Answers:

None of the other answers

Correct answer:

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?

Possible Answers:

Correct answer:

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. 

Possible Answers:

Undefined

Correct answer:

Explanation:

By the property of the sum of logarithms,

.

Example Question #47 : Properties Of Logarithms

Condense the following logarithmic equation:

Possible Answers:

Correct answer:

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

 ?

Possible Answers:

Correct answer:

Explanation:

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

Example Question #49 : Properties Of Logarithms

Write the expression in the most condensed form:

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

Use the Power property of Logarithms:

Rewrite the fractional exponent:

Condense into a fraction using the Quotient property of Logarithms:

Example Question #50 : Properties Of Logarithms

Simplify:  

Possible Answers:

Correct answer:

Explanation:

When logs of the same bases are subtracted, the contents of both logs will be divided with each other.  When logs of the same bases are added, then the contents inside the log will be multiplied together.

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