ACT Math : How to find a logarithm

Study concepts, example questions & explanations for ACT Math

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Example Questions

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Example Question #1 : Logarithms

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

Possible Answers:

1.68794

1.30103

1.39794

1.69897

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

Example Question #1 : How To Find A Logarithm

y = 2x 

If y = 3, approximately what is x?

Round to 4 decimal places.

Possible Answers:

1.8580

2.0000

1.3454

0.6309

1.5850

Correct answer:

1.5850

Explanation:

To solve, we use logarithms. We log both sides and get: log3 = log2x

which can be rewritten as log3 = xlog2

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

Example Question #3 : Logarithms

Evaluate 

log327

Possible Answers:

9

30

27

10

3

Correct answer:

3

Explanation:

You can change the form to 

3x = 27

= 3

Example Question #1 : Logarithms

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

Possible Answers:

7

0

10

24.5

2401

Correct answer:

7

Explanation:

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

\small x^{2}=49

\small x=7

Example Question #1 : Logarithms

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

Possible Answers:

4

16

12

3

2

Correct answer:

4

Explanation:

Given log4= 2, we can determine that 4 to the second power is x; therefore the square root of x is 4.

Example Question #2 : Logarithms

Solve for x in the following equation:

log224 - log23 = logx27

Possible Answers:

3

2

1

2

9

Correct answer:

3

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.

logx27 = 3

x3 = 27

Example Question #1 : Logarithms

What value of  satisfies the equation ?

Possible Answers:

Correct answer:

Explanation:

The answer is .

 can by rewritten as .

In this form the question becomes a simple exponent problem. The answer is  because .

Example Question #3 : Logarithms

If , what is  ?

Possible Answers:

Correct answer:

Explanation:

Use the following equation to easily manipulate all similar logs:

 changes to .

Therefore,  changes to .

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.

Example Question #1 : Logarithms

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

Possible Answers:

2

4

8

32

16

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)

Example Question #2 : Logarithms

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

 

 

Possible Answers:

 

Cannot be simplified into a single logarithm

Correct answer:

Explanation:

Using the property that  , we can simplify the expression to .

Given that  and

We can further simplify this equation to 

 

 

 

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