Solve the Product: log₄6 × log₆9 × log₉4 Logarithm Chain

Logarithm Chain Properties with Cyclic Multiplication

log⁡46×log⁡69×log⁡94= \log_46\times\log_69\times\log_94=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:04 We'll use the formula for multiplication of logarithms
00:10 We'll switch between the bases of the logarithms
00:17 We'll use this formula in our exercise
00:29 We'll use this formula again in the exercise and switch between these bases
00:44 We'll calculate each logarithm separately
00:48 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

log⁡46×log⁡69×log⁡94= \log_46\times\log_69\times\log_94=

2

Step-by-step solution

To solve this problem, we need to recognize that the expression log⁡46×log⁡69×log⁡94\log_46 \times \log_69 \times \log_94 fits the identity of logarithms: log⁡ab×log⁡bc×log⁡ca=1\log_a b \times \log_b c \times \log_c a = 1.

Let us examine the expression:

  • The first term is log⁡46\log_46, where a=4a = 4 and b=6b = 6.
  • The second term is log⁡69\log_69, where b=6b = 6 and c=9c = 9.
  • The third term is log⁡94\log_94, where c=9c = 9 and a=4a = 4.

Notice how log⁡46\log_46, log⁡69\log_69, and log⁡94\log_94 correspond respectively to log⁡ab\log_a b, log⁡bc\log_b c, and log⁡ca\log_c a. Thus, the entire expression matches the multiplication identity of logarithms: log⁡ab×log⁡bc×log⁡ca=1\log_a b \times \log_b c \times \log_c a = 1.

Therefore, the value of the expression log⁡46×log⁡69×log⁡94\log_46 \times \log_69 \times \log_94 is 11.

3

Final Answer

1 1

Key Points to Remember

Essential concepts to master this topic
  • Identity Rule: log⁡ab×log⁡bc×log⁡ca=1 \log_a b \times \log_b c \times \log_c a = 1 for any valid bases
  • Pattern Recognition: Match log⁡46×log⁡69×log⁡94 \log_4 6 \times \log_6 9 \times \log_9 4 to cyclic form
  • Verification: Check that bases cycle: 4→6→9→4 forms complete chain ✓

Common Mistakes

Avoid these frequent errors
  • Computing each logarithm individually then multiplying
    Don't calculate log⁡46≈1.29 \log_4 6 \approx 1.29 , then log⁡69≈1.23 \log_6 9 \approx 1.23 , then multiply = complex decimals and errors! This misses the elegant pattern. Always recognize the cyclic identity log⁡ab×log⁡bc×log⁡ca=1 \log_a b \times \log_b c \times \log_c a = 1 first.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

How do I recognize when to use the cyclic logarithm identity?

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Look for a chain pattern where the argument of one log becomes the base of the next: log⁡46×log⁡69×log⁡94 \log_4 6 \times \log_6 9 \times \log_9 4 . Notice how 6→6, 9→9, and 4→4 cycle back!

Why does this identity always equal 1?

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Using the change of base formula: log⁡ab=ln⁡bln⁡a \log_a b = \frac{\ln b}{\ln a} . When multiplied in a cycle, the numerators and denominators cancel perfectly, leaving 1!

What if the numbers don't form a perfect cycle?

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Then you cannot use this identity! You'd need to calculate each logarithm individually or look for other properties. The cyclic pattern is essential for this shortcut.

Does this work with any three numbers in a cycle?

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Yes! As long as all bases and arguments are positive and not equal to 1, the identity log⁡ab×log⁡bc×log⁡ca=1 \log_a b \times \log_b c \times \log_c a = 1 always holds.

Can I use this identity with more than three logarithms?

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Absolutely! For any cycle like log⁡ab×log⁡bc×log⁡cd×log⁡da \log_a b \times \log_b c \times \log_c d \times \log_d a , the result is still 1. The key is that the chain must close back to the starting base.

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