Solve: 1/4 × log₆(1296) × log₆(1/2) - log₆(3) Expression

Logarithmic Properties with Product-Quotient Rules

14⋅log⁡61296⋅log⁡612−log⁡63= \frac{1}{4}\cdot\log_61296\cdot\log_6\frac{1}{2}-\log_63=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:25 Let's calculate the first logarithm
00:46 This is the solution to the logarithm
01:01 Let's substitute and reduce
01:20 We'll use the formula for subtracting logarithms
01:26 Subtracting logarithms equals the logarithm of the quotient
01:30 Let's use this formula in our exercise
01:53 We'll convert from number to fraction and calculate
01:59 Let's solve the logarithm
02:21 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

14⋅log⁡61296⋅log⁡612−log⁡63= \frac{1}{4}\cdot\log_61296\cdot\log_6\frac{1}{2}-\log_63=

2

Step-by-step solution

We break it down into parts

log⁡61296=x \log_61296=x

6x=1296 6^x=1296

x=4 x=4

14⋅4⋅log⁡612−log⁡63= \frac{1}{4}\cdot4\cdot\log_6\frac{1}{2}-\log_63=

log⁡612−log⁡63= \log_6\frac{1}{2}-\log_63=

log⁡6(12:3)=log⁡616 \log_6\left(\frac{1}{2}:3\right)=\log_6\frac{1}{6}

log⁡616=x \log_6\frac{1}{6}=x

6x=16 6^x=\frac{1}{6}

x=−1 x=-1

3

Final Answer

−1 -1

Key Points to Remember

Essential concepts to master this topic
  • Rule: Use logarithm properties to simplify complex expressions systematically
  • Technique: Convert log⁡6(1296)=4 \log_6(1296) = 4 since 64=1296 6^4 = 1296
  • Check: Verify final answer: log⁡616=−1 \log_6\frac{1}{6} = -1 since 6−1=16 6^{-1} = \frac{1}{6} ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting to apply logarithm properties correctly
    Don't calculate each logarithm separately without using properties = complicated arithmetic! This leads to errors and missed simplifications. Always use properties like log⁡b(x)−log⁡b(y)=log⁡b(xy) \log_b(x) - \log_b(y) = \log_b(\frac{x}{y}) to combine terms 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 find log⁡6(1296) \log_6(1296) without a calculator?

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Think: what power of 6 gives 1296? Try: 61=6 6^1 = 6 , 62=36 6^2 = 36 , 63=216 6^3 = 216 , 64=1296 6^4 = 1296 . So log⁡6(1296)=4 \log_6(1296) = 4 !

Why does log⁡6(12) \log_6(\frac{1}{2}) give a negative result?

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Since 12<1 \frac{1}{2} < 1 , and 6 raised to any positive power is greater than 1, we need a negative exponent. That's why logarithms of numbers between 0 and 1 are always negative.

What's the difference between subtraction and division in logarithms?

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Remember: log⁡b(x)−log⁡b(y)=log⁡b(xy) \log_b(x) - \log_b(y) = \log_b(\frac{x}{y}) . Subtracting logarithms is the same as taking the logarithm of a quotient. This property helps simplify complex expressions!

How do I check if my final answer is correct?

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Substitute back! If your answer is -1, check: 6−1=16 6^{-1} = \frac{1}{6} . Then verify this matches what you get from log⁡6(1/23)=log⁡6(16) \log_6(\frac{1/2}{3}) = \log_6(\frac{1}{6}) .

Why do we multiply by 14 \frac{1}{4} first?

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Follow the order of operations! We calculate 14×4=1 \frac{1}{4} \times 4 = 1 first, which simplifies the expression to just log⁡6(12)−log⁡6(3) \log_6(\frac{1}{2}) - \log_6(3) . This makes the rest much easier!

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