Math Problem Statement

Expand the logarithmic expression \(\log\left(\frac{x^2+1}{x^3(x-1)}\right)\) using the laws of logarithms.

Solution

The provided image contains logarithmic expressions to be expanded using the laws of logarithms. Let's analyze part (c) to identify the issue and correct the expansion. The question requires careful application of the logarithmic properties.

Part (c):

Expression:
log(x2+1x3(x1))\log\left(\frac{x^2 + 1}{x^3(x - 1)}\right)

Correct Steps to Expand:

  1. Quotient Rule:
    log(AB)=log(A)log(B)\log\left(\frac{A}{B}\right) = \log(A) - \log(B)
    Apply to the numerator and denominator: log(x2+1x3(x1))=log(x2+1)log(x3(x1))\log\left(\frac{x^2 + 1}{x^3(x - 1)}\right) = \log(x^2 + 1) - \log\left(x^3(x - 1)\right)

  2. Product Rule for the Denominator:
    log(x3(x1))=log(x3)+log(x1)\log(x^3(x - 1)) = \log(x^3) + \log(x - 1)
    So, the expression becomes: log(x2+1)[log(x3)+log(x1)]\log(x^2 + 1) - \left[\log(x^3) + \log(x - 1)\right]

  3. Power Rule:
    log(xn)=nlog(x)\log(x^n) = n\log(x)
    Apply to the terms with powers: log(x2+1)[3log(x)+log(x1)]\log(x^2 + 1) - \left[3\log(x) + \log(x - 1)\right]

  4. Combine terms: log(x2+1)3log(x)log(x1)\log(x^2 + 1) - 3\log(x) - \log(x - 1)

Final Correct Expansion:

log(x2+1)3log(x)log(x1)\log(x^2 + 1) - 3\log(x) - \log(x - 1)

The error in the given solution was using 15\frac{1}{5} incorrectly. There is no fractional coefficient here, so the expression should match the one derived above.

Would you like to explore any of the logarithmic laws or solve similar problems for clarification?


Related Questions:

  1. What are the key differences between the product, quotient, and power rules in logarithms?
  2. How can you determine when to simplify or expand logarithmic expressions?
  3. What are the common mistakes to avoid when applying the logarithmic laws?
  4. Can you derive a logarithmic equation from a real-world example, such as sound intensity or earthquake magnitudes?
  5. Why is it important to simplify logarithmic expressions in higher-level math, like calculus?

Tip:

Always double-check the application of each logarithmic law step by step to avoid small errors in complex expansions.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra

Formulas

Quotient Rule: \(\log\left(\frac{A}{B}\right) = \log(A) - \log(B)\)
Product Rule: \(\log(AB) = \log(A) + \log(B)\)
Power Rule: \(\log(A^n) = n\log(A)\)

Theorems

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Suitable Grade Level

Grades 10-12