Subtraction of Logs

Subtracting logs involves applying the logarithmic property that allows the expression of the difference of two logs as the log of a quotient.

The subtraction of logs uses the identity: log_b(A) - log_b(B) = log_b(A/B)

This is applicable only when the bases of the logarithms are the same and both A and B are positive numbers.

When subtracting logs, students should follow these steps:

Check the bases: Ensure that the logs have the same base.

Apply the quotient rule: Use the identity log_b(A) - log_b(B) = log_b(A/B) to combine the logs into a single log expression.

Simplify the result: Simplify the log expression if possible, using known log values or further properties.

The following are two methods for subtracting logs:

Method 1: Direct Application of the Quotient Rule 

Step 1: Ensure logs have the same base. 

Step 2: Apply the quotient rule: log_b(A) - log_b(B) = log_b(A/B). 

Step 3: Simplify the resulting expression.

Example: Subtract log_10(100) from log_10(1000).

Solution:

Step 1: Both logs have base 10.

Step 2: log_10(1000) - log_10(100) = log_10(1000/100).

Step 3: Simplifies to log_10(10) = 1.

Method 2: Use of Logarithmic Properties

This method involves using other properties of logarithms, such as change of base, to assist in subtraction.

For example, Subtract ln(20) from ln(100). Solution: ln(100) - ln(20) = ln(100/20) = ln(5).

In logarithms, subtraction has specific properties:

Subtraction follows the quotient rule

The subtraction of logs with the same base can be simplified using the quotient rule: log_b(A) - log_b(B) = log_b(A/B).

Logarithms are not commutative for subtraction Changing the order of the logs changes the result, i.e., log_b(A) - log_b(B) ≠ log_b(B) - log_b(A).

Subtraction is not associative You cannot change the grouping of terms for subtraction: (log_b(A) - log_b(B)) - log_b(C) ≠ log_b(A) - (log_b(B) - log_b(C)).

Subtracting log of 1 Subtracting log_b(1) from log_b(A) leaves the expression unchanged since log_b(1) = 0.

Tips and tricks can help students efficiently handle the subtraction of logs:

Tip 1: Always verify that the logs have the same base before applying the quotient rule.

Tip 2: Simplify the numbers inside the logs first, if possible, to make subtraction easier.

Tip 3: Use known log values, such as log_b(1) = 0, to simplify expressions.

Subtraction of logs can be tricky, leading to common mistakes. Awareness of these errors can help students avoid them.