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ACT Math : How to find a logarithm

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Example Questions

Example Question #1 : Logarithms

Let log 5 = 0.69897 and log 2 = 0.30103. Solve log 50

Possible Answers:

1.69897

1.39794

1.68794

1.30103

1.36903

Correct answer:

1.69897

Explanation:

Using properties of logs:

log (xy) = log x + log y

log (xⁿ) = n log x

log 10 = 1

So log 50 = log (10 * 5) = log 10 + log 5 = 1 + 0.69897 = 1.69897

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Example Question #2 : Logarithms

y = 2^x

If y = 3, approximately what is x?

Round to 4 decimal places.

Possible Answers:

0.6309

1.5850

2.0000

1.8580

1.3454

Correct answer:

1.5850

Explanation:

To solve, we use logarithms. We log both sides and get: log3 = log2^x

which can be rewritten as log3 = xlog2

Then we solve for x: x = log 3/log 2 = 1.5850 . . .

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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:

$7$

$24.5$

$10$

$0$

$2401$

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 #2 : 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 #3 : Logarithms

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

Possible Answers:

$4$

$32$

$8$

$16$

$2$

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 #3 : Logarithms

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

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

Possible Answers:

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

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

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

Cannot be simplified into a single logarithm

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

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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ACT Math : How to find a logarithm

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Logarithms

Example Question #2 : Logarithms

Example Question #3 : Logarithms

Example Question #1 : Logarithms

Example Question #1 : Logarithms

Example Question #2 : Logarithms

Example Question #2 : Logarithms

Example Question #2 : Logarithms

Example Question #3 : Logarithms

Example Question #3 : Logarithms

All ACT Math Resources

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