In these lessons, we will look at four basic rule of logarithms (or properties of logarithms) and how to apply them. You may want to also look at the proofs for these properties.

**Related Pages**

Common And Natural Logarithm

Logarithmic Functions

Rules Of Exponents

Logarithm Properties

The following table gives a summary of the logarithm rules. Scroll down the page for more explanations and examples on how to use the rules to simplify and expand logarithmic expressions.

The rules of logarithms are:

The logarithm of a product is the sum of the logarithms of the factors.

log_{a} xy = log_{a} x + log_{a} y

The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.

log_{a} = log_{a} x – log_{a} y

log_{a} x^{n} = nlog_{a} x

where x and y are positive, and a > 0, a ≠ 1

**Example:**

Simplify the following, expressing each as a single logarithm:

a) log _{2} 4 + log _{2} 5

b) log _{a} 28 – log _{a} 4

c) 2 log _{a} 5 – 3 log _{a} 2

**Solution:**

a) log _{2} 4 + log _{2} 5 = log _{2} (4 × 5) = log _{2} 20

b) log _{a} 28 – log _{a} 4 = log _{a} (28 ÷ 4) = log _{a} 7

c) 2 log _{a} 5 – 3 log _{a} 2 = log _{a} 5^{2} – log _{a} 2^{3} = log _{a}

**Example:**

Evaluate 2 log_{3} 5 + log_{3} 40 – 3 log_{3} 10

**Solution:**

2 log_{3} 5 + log_{3} 40 – 3 log_{3} 10

= log_{3} 5^{2} + log_{3} 40 – log_{3} 10^{3}

= log_{3} 25 + log_{3} 40 – log_{3} 1000

= log_{3}

= log_{3} 1

= 0

**Example:**

Given that log_{2} 3 = 1.585 and log_{2} 5 = 2.322, evaluate log_{4} 15

**Solution:**

- log
_{a}1 = 0 since a^{0}= 1 - log
_{a}a = 1 since a^{1}= a - log
_{a}a^{x}= x since a^{x}= a^{x}

The video explains explains and applies various properties of logarithms. The main focus is how to apply the product, quotient, and power property of logarithms.

**Product property:** The log of a product equals the sum of the logs.

**Quotient Property:** The log of a quotient equals the difference of the logs.

**Power Property:** The log of a power equals the product of the power and the log.

Examples on how to expand logarithmic expression and how to write expressions as a single logarithm.

- Expand the logarithmic expression.

log_{3}(xy^{3}/√z) - Write as a single logarithm.

2 lnx + 1/3 ln(x + 3) - 4 ln(2x)

**Basic Logarithm Properties With Examples**

**Examples:**

Expand the logarithmic expression.

log_{3}(x/5) =

log_{7}(2x) =

log_{5}(x)^{4} =

log_{3}(x/(yz)) =

log_{4}(5√x) =

log_{3}(xy)^{1/2} =

log_{2}((x+1)/(y√z)) =

**How to take an expression involving multiple logarithms and write it as an expression containing only a single logarithm?**

**Example:**

Rewrite as a single logarithm

5 lnx + 13 ln (x^{3} + 5) - 1/2 ln(x + 1)

**Introduction To The First Two Logarithm Properties: Product Law & Quotient Law**

**Property Three And Four Of Logarithms: Power Law & Change Of Base Law**

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