**High School Number Sense Lesson 61: Logarithms**

In previous lessons, we have covered a few different problems concerning exponents. Today's lesson turns the idea of exponents on its head. This concept appeared

**22 times**last year, with a median placement at

**question # 63**.

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The logarithm of a number is

**the exponent**to which another number,

**the base, must be raised**to produce that number. So in logarithmic notation,

**log**

_{B}N = L**B**

^{L}= N**2**

^{4}= 16and

**log**

_{2}16 = 4**Logarithmic Identities**

There are several important formulas concerning logarithms that need to be learned:

**Product**: The logarithm of a**product**is the**sum**of the logarithms of the numbers being multiplied.**Quotient**: The logarithm of the**ratio**of two numbers is the**difference**of the logarithms.**Power**: The logarithm of the**p-th power**of a number is**p times**the logarithm of the number itself.**Root**: The logarithm of a**p-th root**is the logarithm of the number**divided by p**.

**Corresponding Formulas**

**1. Product: log**

2. Quotient: log

3. Power: log

4. Root: log

_{b}(xy) = log_{b}(x) + log_{b}(y)2. Quotient: log

_{b}(x/y) = log_{b}(x) - log_{b}(y)3. Power: log

_{b}(x^{p}) = p log_{b}(x)4. Root: log

_{b}^{p}√x = [log_{b}(x)]/p

**Change of Base**

There is one other formula that will be useful concerning logarithms.**The logarithm log**

_{b}(x) can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:**log**

_{b}(x) = [log_{k}(x)] ÷ [log_{k}(b)]

**Example 1:****If log**

1. Think of this as 4

2. 4

_{4}x = 3, then x = ___1. Think of this as 4

^{3}= x.2. 4

^{3}= 64, so x = 64.**Example 2:**

**If log**

_{16}x = 3/4, then x = ___1. Think of this as 16

^{3/4}= x.

2. First take the 4th root of 16, which is 2.

3. Then take 2 to the 3rd power, which is

**8**.

**Example 3:**

**log**

_{2}8 + log_{2}32 = ___1. Since we are adding two logarithms, we will use the Product Identity. However, it is probably easier to solve each logarithm separately and then add (instead of multiplying 8 by 32).

2. log

_{2}8 =

**3**and log

_{2}32 =

**5**.

3. 3 + 5 =

**8**.

**Example 4:**

**log**6 12 - log

_{6}3 + log_{6}6 = ___

1. This is the same as

**log**= y.

_{6}(3 x 12 ÷ 6)2. Simplify this to

**log**= y.

_{6}63. 6

^{y}= 6, so y =

**1**.

Example 5:

Example 5:

**If log 2 = .3, and log 3 = .5, then log 6 = ___**

1. Notice that 2 x 3 = 6. We are using the Product Identity, so we will add.

2. .3 + .5 =

**.8**

**Example 6:**

**log**

_{9}(27) ÷ log_{9}(3) = ___1. We can change log

_{9}(27) to

**3 log**.

_{9}(3)2. We now have

**3 log**.

_{9}(3) ÷ log_{9}(3)3. We can cancel out the log

_{9}(3), and we are left with

**3 ÷ 1**.

4. The answer is

**3**.

**Check back soon for a free worksheet to help you practice Logarithm.**

**Up Next for High School: PrimeRelative**