Solve the Logarithmic Fraction: log₈5 ÷ log₈9

Change of Base Formula with Logarithmic Division

log⁡85log⁡89= \frac{\log_85}{\log_89}=

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Step-by-step video solution

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00:00 Solve
00:04 We'll use the formula for logical division
00:10 We'll get the log of the numerator in base of the denominator
00:20 We'll use this formula in our exercise
00:28 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

log⁡85log⁡89= \frac{\log_85}{\log_89}=

2

Step-by-step solution

To solve this problem, let's simplify the given expression log⁡85log⁡89\frac{\log_85}{\log_89}.

  • Step 1: Recognize that both the numerator and denominator have the same base, 8.
  • Step 2: The division property of logarithms states that log⁡bMlog⁡bN=log⁡NM\frac{\log_b M}{\log_b N} = \log_N M.
  • Step 3: Apply the division rule to the given expression: log⁡85log⁡89=log⁡95\frac{\log_8 5}{\log_8 9} = \log_9 5.

Thus, after simplifying, we see that log⁡85log⁡89=log⁡95\frac{\log_85}{\log_89} = \log_9 5.

Hence, the correct answer is log⁡95\log_9 5, which corresponds to the choice 1.

3

Final Answer

log⁡95 \log_95

Key Points to Remember

Essential concepts to master this topic
  • Division Property: log⁡bMlog⁡bN=log⁡NM \frac{\log_b M}{\log_b N} = \log_N M changes the base
  • Technique: Transform log⁡85log⁡89 \frac{\log_8 5}{\log_8 9} to log⁡95 \log_9 5
  • Check: Verify denominators become new base: 9 becomes base, 5 stays argument ✓

Common Mistakes

Avoid these frequent errors
  • Applying logarithm properties incorrectly to division
    Don't think log⁡85log⁡89=log⁡859 \frac{\log_8 5}{\log_8 9} = \log_8 \frac{5}{9} ! Division of logarithms doesn't equal logarithm of division. This mixes up quotient rule with change of base. Always use the change of base formula: log⁡bMlog⁡bN=log⁡NM \frac{\log_b M}{\log_b N} = \log_N M .

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why doesn't the division of logarithms equal the logarithm of the division?

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Because logarithm properties work differently for division! The quotient rule log⁡bMN=log⁡bM−log⁡bN \log_b \frac{M}{N} = \log_b M - \log_b N is for subtraction, not division of logs. When dividing logarithms with the same base, use the change of base formula instead.

How do I remember which number becomes the new base?

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In log⁡bMlog⁡bN=log⁡NM \frac{\log_b M}{\log_b N} = \log_N M , the denominator becomes the new base and the numerator becomes the argument. Think: "bottom becomes base, top stays on top!"

Can I use this formula with any logarithm base?

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Yes! The change of base division formula works with any base as long as both logarithms in the fraction have the same original base. It could be base 10, base e, or any other positive number.

What if I get confused about which property to use?

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Look at the structure! If you see division of two logarithms with the same base, use change of base. If you see logarithm of a fraction, use the quotient rule for subtraction.

How can I check my answer is correct?

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Convert both sides back to the original base using the change of base formula! log⁡95=log⁡85log⁡89 \log_9 5 = \frac{\log_8 5}{\log_8 9} , which matches our original expression ✓

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