# ACT Math : Logarithms

## Example Questions

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### Example Question #1 : Logarithms

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

1.36903

1.39794

1.30103

1.69897

1.68794

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 : Logarithms

y = 2x

If y = 3, approximately what is x?

Round to 4 decimal places.

1.5850

0.6309

2.0000

1.8580

1.3454

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 : How To Find A Logarithm

Evaluate

log327

9

3

30

10

27

3

Explanation:

You can change the form to

3x = 27

= 3

If , what is ?

Explanation:

If , then

### Example Question #5 : How To Find A Logarithm

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

2

3

4

12

16

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 #1 : Logarithms

Solve for x in the following equation:

log224 - log23 = logx27

2

9

3

2

1

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 ?

Explanation:

can by rewritten as .

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

### Example Question #8 : How To Find A Logarithm

If , what is  ?

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  that satisfies  ?

Explanation:

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?

Cannot be simplified into a single logarithm

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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