Algebra II : Solving and Graphing Logarithms

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #164 : Logarithms

Solve the equation

Possible Answers:

No Solution

Correct answer:

No Solution

Explanation:

Set the arguments of the natural log equal and solve for 

However, you cannot take the natural log of a negative number. Therefore; there is No Solution

Example Question #165 : Logarithms

Solve:

Possible Answers:

Correct answer:

Explanation:

To solve this problem, it is easiest to start by thinking about what the logarithm means:

Now, we take the log of both sides (either base-10 or natural):

Next, using the properties of logarithms, we can bring the exponent x in front of the logarithm:

Finally, solve for x:

Example Question #166 : Logarithms

Evaluate

Possible Answers:

Correct answer:

Explanation:

Logarithms are inverses of exponents. 

 is asking how many fives multiplied together are equal to 25.

The formula to solve logarithmic equations is as follows.

Applying this formula to our particular problem results in the following.

 

 = 

So .

Example Question #167 : Logarithms

Solve:

Possible Answers:

Correct answer:

Explanation:

To solve, we must keep in mind that a logarithm is asking the following:

means , and we must find x.

First, we can rewrite the sum of the base-8 logarithms as a product (this is a property of logarithms):

Now, we use the above formula to find what this numerically equals:

Take the natural or common logarithm of both sides, which allows us to bring the power x in front of the logarithm:

Example Question #168 : Logarithms

Solve for x:

Possible Answers:

Correct answer:

Explanation:

To solve for x, we must take the logarithm of both sides (common or natural, it doesn't matter):

In doing this, we can now bring the exponents in front of the logarithms:

Now, solve for x:

Example Question #169 : Logarithms

If  and , what is ?

Possible Answers:

Correct answer:

Explanation:

First, we know from logarithmic properties that:

So if we can find a combination of our knowns that equal , we should be able to figure out .  It may take some guess and check, but:

That would equate to:

Using our log properties, we can simplify into:

That means that:

 

Example Question #170 : Logarithms

Evaluate .

Possible Answers:

Correct answer:

Explanation:

There are a few ways to go about this, but let's use a change of base to make the problem easier to work without using a calculator.  First, we know that:

We can choose any value for  that we would like here, so to make things simple, let's put this problem in base :

 

Example Question #21 : Solving Logarithms

Evaluate .

Possible Answers:

Correct answer:

Explanation:

We can take this problem and expand it a bit, which will make things easier in the long run.  We know that:

Using one of our logarithmic properties, we can expand even further:

Another log property states that:

So:

Example Question #22 : Solving Logarithms

Solve:

Possible Answers:

Correct answer:

Explanation:

To solve for x, remember that exponents inside logarithms can be moved to the front of the logarithm:

Next, we can rewrite the logarithm as the number it equals, and solve for x:

Example Question #23 : Solving Logarithms

Solve:  

Possible Answers:

Correct answer:

Explanation:

In order to solve for the x-variable, we will need to exponential function both sides of the equation in order to eliminate the natural log.

The equation becomes:

Divide by two on both sides.

The answer is:  

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