ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #2292 : Act Math

Solve the following equation

.

Possible Answers:

Correct answer:

Explanation:

In order to solve a question like this, you will need to use logarithms. First, start by converting this into a basic logarithm:

Recall that you need to convert  for your calculator:

, which equals approximately 

Thus, you can solve for :

Example Question #2293 : Act Math

At the end of each year, an account compounds interest at a rate of .  If the account began with , how many years will it take for it to reach a value of , presuming no withdrawals or deposits occur?

Possible Answers:

Correct answer:

Explanation:

The general function that defines this compounding interest is:

, where  is the number of years.

What we are looking for is:

You can solve this using a logarithm.  First, isolate the variable term by dividing both sides:

Which is:

Next, recall that this is the logarithm:

For this, you will need to do a base conversion:

This is 

This means that it will take  years.   is too few and at the end of , you will have over .

Example Question #2294 : Act Math

What is the value of ?  Round to the nearest hundredth.

Possible Answers:

Correct answer:

Explanation:

Remember that you will need to calculate your logarithm by doing a base conversion. This is done by changing  into:

Using your calculator, you can find this to be:

 or approximately 

Example Question #11 : How To Find A Logarithm

if , what is ?

Possible Answers:

Correct answer:

Explanation:

The first step of this problem is to find

 by expanding to the formula

  

y is found to be 2.  The next step is to plug y in to the second log.  

, which expands to

Example Question #101 : Exponents

Find  .

Possible Answers:

Correct answer:

Explanation:

 

expands to

 

expands to

Example Question #11 : Logarithms

Simplify:

 

Possible Answers:

Correct answer:

Explanation:

Here, we need to make use of some logarithm identities:    

Therefore, putting all of those things together, we get the final answer of 

Example Question #11 : Logarithms

If

,

then what is ?

Possible Answers:

Correct answer:

Explanation:

This is a test of translating logarithmic/exponential properties, with the key here being to realize that

is equivalent to .

With that in mind, here is how it works out:

Hence, .

Example Question #12 : Logarithms

can be written as which of the following?

A.

B.

C.

Possible Answers:

B and C only

A only

B only

A, B and C

A and B only

Correct answer:

A, B and C

Explanation:

A is true in two ways. You can use the fact that if a logarithm has no base, you can use base 10, or you can use the fact that you can use this property:

B is a simple change of base application, and C is simply computing the logarithm.

Example Question #801 : Algebra

If , then  ?

Possible Answers:

10

4

5

25

15

Correct answer:

5

Explanation:

 

Calculate the power of  that makes the expression equal to 25. We can set up an alternate or equivalent equation to solve this problem:

Solve this equation by taking the square root of both sides. 

, because logarithmic equations cannot have a negative base.

The solution to this expression is:

 

Example Question #802 : Algebra

If  and , then which of the following CANNOT be the value of ?

 

Possible Answers:

Correct answer:

Explanation:

Even roots of numbers can either be positive or negative. Thus, x = +/- 5 and y = +/- 3. The possible values from x + y can therefore be:

(-5) + (-3) = -8

(-5) + 3 = -2

5 + (-3) = 2

5 + 3 = 8

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