Solve log₂9 - log₂3: Base-2 Logarithm Subtraction Problem

Logarithm Properties with Subtraction Rule

log⁡29−log⁡23= \log_29-\log_23=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:05 We will use the formula for subtracting logarithms
00:19 Subtracting logarithms equals the logarithm of the quotient of numbers
00:31 We will use this formula in our exercise
00:42 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

log⁡29−log⁡23= \log_29-\log_23=

2

Step-by-step solution

To solve the problem of evaluating log⁡29−log⁡23\log_2 9 - \log_2 3, we apply the properties of logarithms as follows:

  • Step 1: Recognize that the expression uses a subtraction of logarithms with the same base: log⁡29−log⁡23\log_2 9 - \log_2 3.
  • Step 2: Use the logarithmic subtraction rule: log⁡bA−log⁡bB=log⁡b(AB)\log_b A - \log_b B = \log_b \left(\frac{A}{B}\right).
  • Step 3: Simplify using this rule: log⁡29−log⁡23=log⁡2(93)\log_2 9 - \log_2 3 = \log_2 \left(\frac{9}{3}\right).
  • Step 4: Perform the division: 93=3\frac{9}{3} = 3.
  • Step 5: Therefore, log⁡2(93)=log⁡23\log_2 \left(\frac{9}{3}\right) = \log_2 3.

Thus, the simplified and evaluated result is log⁡23 \log_2 3 .

3

Final Answer

log⁡23 \log_23

Key Points to Remember

Essential concepts to master this topic
  • Rule: log⁡bA−log⁡bB=log⁡b(AB) \log_b A - \log_b B = \log_b \left(\frac{A}{B}\right)
  • Technique: Apply subtraction rule: log⁡29−log⁡23=log⁡2(93) \log_2 9 - \log_2 3 = \log_2 \left(\frac{9}{3}\right)
  • Check: Verify division: 93=3 \frac{9}{3} = 3 , so answer is log⁡23 \log_2 3 ✓

Common Mistakes

Avoid these frequent errors
  • Adding or subtracting the numbers inside the logarithms
    Don't try log⁡2(9−3)=log⁡26 \log_2(9-3) = \log_2 6 ! This gives the wrong answer because logarithm subtraction doesn't work like regular arithmetic. Always use the quotient rule: log⁡bA−log⁡bB=log⁡b(AB) \log_b A - \log_b B = \log_b \left(\frac{A}{B}\right) .

Practice Quiz

Test your knowledge with interactive questions

\( \log_{10}3+\log_{10}4= \)

FAQ

Everything you need to know about this question

Why can't I just subtract 9 - 3 = 6 inside the logarithm?

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Because logarithms follow special rules, not regular arithmetic! log⁡29−log⁡23 \log_2 9 - \log_2 3 means you're subtracting two separate logarithms, which requires the quotient rule, not simple subtraction.

How do I remember the logarithm subtraction rule?

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Think "subtraction becomes division"! When you subtract logarithms with the same base, you divide what's inside: log⁡bA−log⁡bB=log⁡b(AB) \log_b A - \log_b B = \log_b \left(\frac{A}{B}\right) .

What if the bases are different?

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You cannot use the subtraction rule when bases are different! For example, log⁡29−log⁡33 \log_2 9 - \log_3 3 cannot be simplified using this rule. The bases must match.

Do I need to calculate the actual value of log₂3?

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Not necessarily! The answer log⁡23 \log_2 3 is already in simplest form. Unless specifically asked for a decimal approximation, leave it as log⁡23 \log_2 3 .

Can I check my answer somehow?

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Yes! Use the definition of logarithms: if log⁡23=x \log_2 3 = x , then 2x=3 2^x = 3 . You can verify that log⁡29−log⁡23 \log_2 9 - \log_2 3 equals this same value.

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