Logarithm Rules

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.

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The rules of logarithms are:

1) Product Rule

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

log_a xy = log_a x + log_a y

2) Quotient Rule

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

3) Power Rule

log_a xⁿ = nlog_a x

4) Change Of Base Rule

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 5² – log _a 2³ = log _a

Example:
Evaluate 2 log₃ 5 + log₃ 40 – 3 log₃ 10

Solution:
2 log₃ 5 + log₃ 40 – 3 log₃ 10
= log₃ 5² + log₃ 40 – log₃ 10³
= log₃ 25 + log₃ 40 – log₃ 1000
= log₃
= log₃ 1
= 0

Example:
Given that log₂ 3 = 1.585 and log₂ 5 = 2.322, evaluate log₄ 15

Solution:

The Properties Of Logarithms And Applications

1. log_a 1 = 0 since a⁰ = 1
2. log_a a = 1 since a¹ = a
3. 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.

1. Expand the logarithmic expression.
   log₃(xy³/√z)
2. Write as a single logarithm.
   2 lnx + 1/3 ln(x + 3) - 4 ln(2x)

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Basic Logarithm Properties With Examples

Examples:
Expand the logarithmic expression.
log₃(x/5) =
log₇(2x) =
log₅(x)⁴ =
log₃(x/(yz)) =
log₄(5√x) =
log₃(xy)^1/2 =
log₂((x+1)/(y√z)) =

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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 (x³ + 5) - 1/2 ln(x + 1)

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

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

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