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ACT Math : Algebra

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Example Questions

Example Question #3 : Logarithms

Evaluate

log₃27

Possible Answers:

Correct answer:

Explanation:

You can change the form to

3^x= 27

x = 3

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Example Question #1 : Logarithms

If $\small \log_{x}49=2$ , what is $\small x$ ?

Possible Answers:

$2401$

$24.5$

$10$

$0$

$7$

Correct answer:

$7$

Explanation:

If $\small \log_{x}y=z$ , then $\small x^{z}=y$

$\small x^{2}=49$

$\small x=7$

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Example Question #1 : Logarithms

If log₄ x = 2, what is the square root of x?

Possible Answers:

Correct answer:

Explanation:

Given log₄x = 2, we can determine that 4 to the second power is x; therefore the square root of x is 4.

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Example Question #2 : Logarithms

Solve for x in the following equation:

log₂24 - log₂3 = log_x27

Possible Answers:

–2

Correct answer:

Explanation:

Since the two logarithmic expressions on the left side of the equation have the same base, you can use the quotient rule to re-express them as the following:

log₂24 – log₂3 = log₂(24/3) = log₂8 = 3

Therefore we have the following equivalent expressions, from which it can be deduced that x = 3.

log_x27 = 3

x³ = 27

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Example Question #2 : Logarithms

What value of satisfies the equation $\log _{x}64=2$ ?

Possible Answers:

Correct answer:

Explanation:

The answer is .

$\log _{x}64=2$ can by rewritten as $x^{2}=64$ .

In this form the question becomes a simple exponent problem. The answer is because $8^{2}=64$ .

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Example Question #4 : Logarithms

If $\log _{2}64=x$ , what is ?

Possible Answers:

Correct answer:

Explanation:

Use the following equation to easily manipulate all similar logs:

$\log _{a}b=x$ changes to $a^{x}=b$ .

Therefore, $\log _{2}64=x$ changes to $2^{x}=64$ .

2 raised to the power of 6 yields 64, so must equal 6. If finding the 6 was difficult from the formula, simply keep multiplying 2 by itself until you reach 64.

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Example Question #5 : Logarithms

Which of the following is a value of $x$ that satisfies $\log_{x}64=2$ ?

Possible Answers:

$2$

$4$

$16$

$32$

$8$

Correct answer:

$8$

Explanation:

The general equation of a logarithm is $\log_{x}a=b$ , and $x^{b}=a$

In this case, $x^{2}=64$ , and thus $x=8$ (or $-8$ , but $-8$ is not an answer choice)

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Example Question #791 : Algebra

How can we simplify this expression below into a single logarithm?

$3log(x)+\frac{1}{2}log(4y)-log(z)$

Possible Answers:

$log(\frac{4x^{3}y^{\frac{1}{2}}}{z})$

$log(2x^{3}+y^{\frac{1}{2}}-z)$

Cannot be simplified into a single logarithm

$log(\frac{2x^{3}y^{\frac{1}{2}}}{z})$

$\frac{3}{2}log(\frac{4xy}{z})$

Correct answer:

$log(\frac{2x^{3}y^{\frac{1}{2}}}{z})$

Explanation:

Using the property that $log(x^{n})=nlog(x)$ , we can simplify the expression to $log(x^{3})+log((4y)^{\frac{1}{2}})-log(z)$ .

Given that log(xy)=log(x)+log(y) and $log(\frac{x}{y})=log(x)-log(y)$

We can further simplify this equation to $log(\frac{2x^{3}y^{\frac{1}{2}}}{z})$

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Example Question #101 : Exponents

What is the value of $\textup{log}_2(64)$ ? Round to the nearest hundredth.

Possible Answers:

4.5

1.81

4.16

Correct answer:

Explanation:

You could solve this by using your calculator. Remember that you will have to translate this into:

$\frac{log(64)}{log(2)}$

Another way you can solve it is by noticing that 64=2^6

This means you can rewrite your logarithm:

log_2(2^6)

Applying logarithm rules, you can factor out the power:

log_2(2^6)=6log_2(2)

For any value , log_n(n) = 1 . Therefore, log_2(2) = 1 . So, your answer is .

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Example Question #2291 : Act Math

Solve for

5^x^+^1=60 .

Round to the nearest hundredth.

Possible Answers:

1.75

1.54

3.73

2.54

4.73

Correct answer:

1.54

Explanation:

To solve an exponential equation like this, you need to use logarithms. This can be translated into:

log_5(60)=x+1

Now, remember that your calculator needs to have this translated. The logarithm log_5(60) is equal to the following:

$\frac{log(60)}{log(5)}$ , which equals approximately 2.54 .

Remember that you have the equation:

2.54 = x + 1

Thus, x=1.54 .

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ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #3 : Logarithms

Example Question #1 : Logarithms

Example Question #1 : Logarithms

Example Question #2 : Logarithms

Example Question #2 : Logarithms

Example Question #4 : Logarithms

Example Question #5 : Logarithms

Example Question #791 : Algebra

Example Question #101 : Exponents

Example Question #2291 : Act Math

All ACT Math Resources

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