The Logarithm Quotient Rule: Simplifying Expressions

In the realm of mathematics, logarithms play a crucial role in simplifying complex expressions and solving equations. One of the fundamental rules governing logarithms is the quotient rule, which allows us to express the logarithm of a quotient in terms of the logarithms of the individual terms. This rule is particularly useful when dealing with expressions involving division.

Understanding the Logarithm Quotient Rule

The logarithm quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. Mathematically, this can be expressed as:

log_b(x/y) = log_b(x) - log_b(y)

where:

• b is the base of the logarithm (b > 0 and b ≠ 1)
• x and y are positive real numbers

Derivation of the Logarithm Quotient Rule

To understand the derivation of this rule, let's start with the definition of a logarithm:

log_b(x) = y if and only if b^y = x

Now, consider the quotient x/y. Let's express this quotient in logarithmic form:

log_b(x/y) = z

This means:

b^z = x/y

Multiplying both sides by y, we get:

b^zy = x

Now, let's take the logarithm of both sides with base b:

log_b(b^zy) = log_b(x)

Using the logarithm product rule (log_b(uv) = log_b(u) + log_b(v)), we can simplify the left-hand side:

log_b(b^z) + log_b(y) = log_b(x)

Since log_b(b^z) = z, we have:

z + log_b(y) = log_b(x)

Subtracting log_b(y) from both sides, we arrive at:

z = log_b(x) - log_b(y)

Since we defined z = log_b(x/y), we can substitute it back into the equation:

log_b(x/y) = log_b(x) - log_b(y)

This concludes the derivation of the logarithm quotient rule.

Examples of the Logarithm Quotient Rule

Let's illustrate the application of the logarithm quotient rule with some examples:

Example 1:

Simplify the expression log₂(8/4).

Using the quotient rule:

log₂(8/4) = log₂(8) - log₂(4)

We know that log₂(8) = 3 and log₂(4) = 2, so:

log₂(8/4) = 3 - 2 = 1

Example 2:

Simplify the expression log₁₀(100/10).

Applying the quotient rule:

log₁₀(100/10) = log₁₀(100) - log₁₀(10)

Since log₁₀(100) = 2 and log₁₀(10) = 1, we have:

log₁₀(100/10) = 2 - 1 = 1

Conclusion

The logarithm quotient rule is a powerful tool for simplifying expressions involving division. By understanding this rule and its derivation, we can effectively manipulate logarithmic expressions and solve mathematical problems involving quotients.