How to find a logarithm - ACT Math

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Question

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

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

Answer

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 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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Question

If $\log _{x}{25}=2$ , then ?

Answer

$\log _{x}{25}=2$

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.

$\sqrt{25}=\sqrt{x^2}$

$x=\pm 5$

$x\neq - 5$ , because logarithmic equations cannot have a negative base.

The solution to this expression is:

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Question

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

Answer

Given log4_x_ = 2, we can determine that 4 to the second power is x; therefore the square root of x is 4.

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Question

Solve for x in the following equation:

log224 - log23 = log_x_27

Answer

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_x_27 = 3

_x_3 = 27

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Question

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

Answer

Using properties of logs:

log (xy) = log x + log y

log (_x_n) = 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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Question

y = 2x

If y = 3, approximately what is x?

Round to 4 decimal places.

Answer

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 . . .

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Question

Evaluate

log327

Answer

You can change the form to

3_x_ = 27

x = 3

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Question

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

Answer

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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Question

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

Answer

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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Question

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

Answer

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

$\small x^{2}=49$

$\small x=7$

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Question

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

Answer

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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Question

If

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

then what is ?

Answer

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:

$4^{5/2} = (\sqrt[2]{4})^{5} = 2^{5} = 22222 = 32$

Hence, .

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Question

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

Answer

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

This means you can rewrite your logarithm:

Applying logarithm rules, you can factor out the power:

For any value , . Therefore, . So, your answer is .

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Question

Solve for

.

Round to the nearest hundredth.

Answer

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

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

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

Remember that you have the equation:

Thus, .

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Question

Solve the following equation

.

Answer

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:

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

Thus, you can solve for :

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Question

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

Answer

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

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

Using your calculator, you can find this to be:

or approximately

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Question

At the end of each year, an account compounds interest at a rate of . 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?

Answer

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:

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

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

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

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Question

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

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

B. $\log_{5}25$

C.

Answer

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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Question

if , what is ?

Answer

The first step of this problem is to find

$\log_4(16)$ 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

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Question

Find .

Answer

expands to

expands to

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