### All ACT Math Resources

## Example Questions

### Example Question #1 : Logarithms

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

**Possible Answers:**

1.30103

1.36903

1.39794

1.68794

1.69897

**Correct answer:**

1.69897

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

### Example Question #1 : How To Find A Logarithm

y = 2^{x}

If y = 3, approximately what is x?

Round to 4 decimal places.

**Possible Answers:**

0.6309

1.8580

2.0000

1.5850

1.3454

**Correct answer:**

1.5850

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 . . .

### Example Question #1 : Logarithms

Evaluate

log_{3}27

**Possible Answers:**

3

10

9

27

30

**Correct answer:**

3

You can change the form to

3^{x }= 27

*x *= 3

### Example Question #4 : Logarithms

If , what is ?

**Possible Answers:**

**Correct answer:**

If , then

### Example Question #2 : Logarithms

If log_{4 } x = 2, what is the square root of x?

**Possible Answers:**

12

4

2

3

16

**Correct answer:**

4

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

### Example Question #1 : How To Find A Logarithm

Solve for x in the following equation:

log_{2}24 - log_{2}3 = log* _{x}*27

**Possible Answers:**

**–**2

1

2

3

9

**Correct answer:**

3

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:

log_{2}24 **–** log_{2}3 = log_{2}(24/3) = log_{2}8 = 3

Therefore we have the following equivalent expressions, from which it can be deduced that *x *= 3.

log* _{x}*27 = 3

*x*^{3} = 27

### Example Question #1 : Logarithms

What value of satisfies the equation ?

**Possible Answers:**

**Correct answer:**

The answer is .

can by rewritten as .

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

### Example Question #8 : Logarithms

If , what is ?

**Possible Answers:**

**Correct answer:**

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 #9 : Logarithms

Which of the following is a value of that satisfies ?

**Possible Answers:**

**Correct answer:**

The general equation of a logarithm is , and

In this case, , and thus (or , but is not an answer choice)

### Example Question #1 : Logarithms

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

**Possible Answers:**

Cannot be simplified into a single logarithm

**Correct answer:**

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

Given that and

We can further simplify this equation to