Solve the Logarithm Equation: x∙log_m(1/3^x)

Logarithm Multiplication with Power Expressions

xlog⁡m13x= x\log_m\frac{1}{3^x}=

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Step-by-step video solution

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00:00 Solve
00:04 We will use the formula to convert a fraction to a negative exponent
00:09 We will use this formula in our exercise
00:19 We will use the formula for the logarithm of a power
00:24 We will use this formula in our exercise
00:36 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

xlog⁡m13x= x\log_m\frac{1}{3^x}=

2

Step-by-step solution

To solve this problem, we will apply the rules of logarithms as follows:

  • Firstly, rewrite the expression log⁡m13x \log_m \frac{1}{3^x} using the Quotient Rule:

    • log⁡m13x=log⁡m1−log⁡m3x \log_m \frac{1}{3^x} = \log_m 1 - \log_m 3^x

  • Since log⁡m1=0 \log_m 1 = 0 , the expression simplifies to:

    • 0−log⁡m3x=−log⁡m3x 0 - \log_m 3^x = -\log_m 3^x

  • Apply the Power Rule to simplify −log⁡m3x-\log_m 3^x:

    • −log⁡m3x=−xlog⁡m3 -\log_m 3^x = -x \log_m 3

  • Substitute back to the original expression xlog⁡m13x x \log_m \frac{1}{3^x} :

    • x(−xlog⁡m3)=−x2log⁡m3 x ( -x \log_m 3) = -x^2 \log_m 3

Therefore, the solution to the problem in terms of simplifying the expression is −x2log⁡m3 -x^2 \log_m 3 .

3

Final Answer

−x2log⁡m3 -x^2\log_m3

Key Points to Remember

Essential concepts to master this topic
  • Rule: Use logarithm properties to simplify fractions and powers
  • Technique: Convert 13x \frac{1}{3^x} to −xlog⁡m3 -x \log_m 3 using quotient and power rules
  • Check: Verify final form matches answer choices: −x2log⁡m3 -x^2 \log_m 3 ✓

Common Mistakes

Avoid these frequent errors
  • Incorrectly applying logarithm rules to the coefficient x
    Don't treat the coefficient x as part of the logarithm argument = wrong distribution! This gives expressions like log⁡xm \log_{xm} which is incorrect. Always apply logarithm properties only to the argument inside the log, then multiply by external coefficients.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why does log⁡m13x \log_m \frac{1}{3^x} become negative?

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Because fractions with 1 in the numerator create negative logarithms! log⁡m13x=log⁡m1−log⁡m3x=0−xlog⁡m3=−xlog⁡m3 \log_m \frac{1}{3^x} = \log_m 1 - \log_m 3^x = 0 - x \log_m 3 = -x \log_m 3

How do I remember the quotient rule for logarithms?

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Think "log of a fraction equals log of top minus log of bottom": log⁡bAB=log⁡bA−log⁡bB \log_b \frac{A}{B} = \log_b A - \log_b B . It's like division becomes subtraction in logarithms!

Why does the x move outside as x2 x^2 ?

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We have two x's multiplying: the coefficient x times the x from the power rule! x⋅(−xlog⁡m3)=−x2log⁡m3 x \cdot (-x \log_m 3) = -x^2 \log_m 3 . Always multiply coefficients carefully.

What if the base m equals 3?

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Then log⁡m3=log⁡33=1 \log_m 3 = \log_3 3 = 1 , so your final answer becomes simply −x2 -x^2 ! The logarithm of a number to its own base always equals 1.

Can I solve this without using logarithm properties?

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No, you must use logarithm properties! This expression cannot be simplified further without applying the quotient rule and power rule for logarithms. These properties are essential tools.

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