Logarithm

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_BN = L

is equivalent to:

B^L = N

For example, these two notations are equivalent:

2⁴ = 16
and
log₂16 = 4

Logarithmic Identities

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

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

Corresponding Formulas

1. Product: 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 log₄ x = 3, then x = ___
1. Think of this as 4³ = x.
2. 4³ = 64, so x = 64.

Example 2:

If log₁₆ 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:

log₂ 8 + log₂ 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. log₂ 8 = 3 and log₂ 32 = 5.
3. 3 + 5 = 8.

Example 4:

log₆ 3 + log6 12 - log₆ 6 = ___
1. This is the same as log₆ (3 x 12 ÷ 6) = y.
2. Simplify this to log₆ 6 = y.
3. 6^y = 6, so y = 1.

​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:

log₉ (27) ÷ log₉ (3) = ___
1. We can change log₉ (27) to 3 log₉ (3).
2. We now have 3 log₉ (3) ÷ log₉ (3).
3. We can cancel out the log₉ (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