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ACT Math : How to find a logarithm

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Example Questions

Example Question #11 : Logarithms

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

Possible Answers:

4.16

1.81

4.5

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 #12 : Logarithms

Solve for

5^x^+^1=60 .

Round to the nearest hundredth.

Possible Answers:

3.73

4.73

1.54

2.54

1.75

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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Example Question #13 : Logarithms

Solve the following equation

3^2^x^-^4=44 .

Possible Answers:

7.31

3.44

3.72

2.11

Correct answer:

3.72

Explanation:

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

log_3(44)=2x-4

Recall that you need to convert log_3(44) for your calculator:

$\frac{log(44)}{log(3)}$ , which equals approximately 3.44

Thus, you can solve for :

3.44=2x-4

2x = 7.44

x=3.72

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Example Question #2293 : Act Math

At the end of each year, an account compounds interest at a rate of $4.5\%$ . If the account began with $\$1400$ , how many years will it take for it to reach a value of $\$5000$ , presuming no withdrawals or deposits occur?

Possible Answers:

Correct answer:

Explanation:

The general function that defines this compounding interest is:

value = 1.045^t* 1400 , where is the number of years.

What we are looking for is:

1.045^t* 1400 = 5000

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

$1.045^t =\frac{5000}{1400}$

Which is:

$1.045^t =\frac{25}{7}$

Next, recall that this is the logarithm:

$t = log_1_._0_4_5(\frac{25}{7})$

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

$log_1_._0_4_5(\frac{25}{7}) = \frac{log(\frac{25}{7})}{log(1.045)}$

This is 28.9199397858296...

This means that it will take years. is too few and at the end of , you will have over 5000 .

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Example Question #2294 : Act Math

What is the value of $\textup{log}_4_._5(381)$ ? Round to the nearest hundredth.

Possible Answers:

3.95

2.58

5.94

8.14

4.31

Correct answer:

3.95

Explanation:

Remember that you will need to calculate your logarithm by doing a base conversion. This is done by changing log_4_._5(381) into:

$\frac{log(381)}{log(4.5)}$

Using your calculator, you can find this to be:

3.95112604435386... or approximately 3.95

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Example Question #101 : Exponents

if log_4(16) = x , what is log_2(x) ?

Possible Answers:

x = 4

x = 1

x = 2

x = 1/2

x = 0

Correct answer:

x = 1

Explanation:

The first step of this problem is to find

$\log_4(16)$ by expanding to the formula

4^y = 16.

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

log_2(2) , which expands to

2^x = 2,

x = 1

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Example Question #102 : Exponents

Find log_2(log_3(81)) .

Possible Answers:

x = 1/3

x = 1/2

x = 0

x = 2

x = 1

Correct answer:

x = 2

Explanation:

log_3(81)

expands to

3^y = 81, y = 4

log_2(4)

expands to

2^x = 4, x = 2

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Example Question #11 : Logarithms

Simplify:

$\textup{log a}^2+\textup{log b}+\textup{log c}^3.$

Possible Answers:

$\textup{log}\left (a^2bc^3 \right )$

$\textup{log}\left (\frac{a^2b}{c} \right )$

$\textup{log}\left (\frac{2ab}{3c} \right )$

$\textup{log}\left (\frac{ab}{c} \right )$

$\textup{log}\left (6abc \right )$

Correct answer:

$\textup{log}\left (\frac{2ab}{3c} \right )$

Explanation:

Here, we need to make use of some logarithm identities: $\textup{log}(\textup{a}^\textup{n})=\textup{n}*\textup{log(a)}log(a)+\textup{log}(\textup{b})=\textup{log(ab)}$

$\textup{and } \textup{log(a)}-\textup{log(b)}=log(\frac{a}{b}).$

Therefore, putting all of those things together, we get the final answer of $\textup{log a}^2+\textup{log b}+\textup{log c}^3= \textup{log}(\frac{2\textup{ab}}{3\textup{c}}).$

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Example Question #11 : Logarithms

$\log_{4}x = \frac{5}{2}$ ,

then what is ?

Possible Answers:

128

Correct answer:

Explanation:

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

$\log_{4}x = \frac{5}{2}$ is equivalent to $x = 4^{5/2}$ .

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

$4^{5/2} = (\sqrt[2]{4})^{5} = 2^{5} = 2*2*2*2*2 = 32$

Hence, x = 32 .

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Example Question #12 : Logarithms

$\frac{\log{25}}{\log{5}}$ can be written as which of the following?

A. $\frac{\log_{10}25}{\log_{10}5}$

B. $\log_{5}25$

Possible Answers:

A, B and C

A and B only

A only

B only

B and C 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:

$\log_{a}b = \frac{\log_{c}b}{\log_{c}a}$

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

$\log_{5}25 = \frac{\log_{10}25}{\log_{10}5}$

$\log_5 25\rightarrow \log_ab=c \rightarrow a^c=b$

$\\ \log_5 25\\ 5^x=25\\5^2=25\\x=2$

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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 #11 : Logarithms

Example Question #12 : Logarithms

Example Question #13 : Logarithms

Example Question #2293 : Act Math

Example Question #2294 : Act Math

Example Question #101 : Exponents

Example Question #102 : Exponents

Example Question #11 : Logarithms

Example Question #11 : Logarithms

Example Question #12 : Logarithms

All ACT Math Resources

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