Test: Logarithms - ACT MCQ

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

You can change the form to 

3^x = 27

x = 3

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

The answer is 8.

log_x 64=2 can by rewritten as x² = 64.

In this form the question becomes a simple exponent problem. The answer is 8 because 8² = 64.

Remember that you will need to calculate your logarithm by doing a base conversion. This is done by changing log_4.5(381) into:

Using your calculator, you can find this to be:

3.95112604435386... or approximately 3.95

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:

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

log_x 25 = 2
Calculate the power of x that makes the expression equal to 25. We can set up an alternate or equivalent equation to solve this problem:
25 = x²
Solve this equation by taking the square root of both sides. 
√25 = √x²
x = ± 5
x ≠ −5, because logarithmic equations cannot have a negative base.
The solution to this expression is:
x = 5 

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

log₅(60) = x + 1

Now, remember that your calculator needs to have this translated.  The logarithm log₅(60) is equal to the following:

which equals approximately 2.54.

Remember that you have the equation:

2.54 = x + 1

Thus, x = 1.54.

The first step of this problem is to find

log₄(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.  

log₂(2), which expands to

2^x = 2,

x = 1

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