Math Problem Statement

Simplify the logarithmic expression log(√(x³)√[³](y²) / z⁴).

Solution

The expression in the image is:

log(x3y23z4)\log \left( \frac{\sqrt{x^3} \cdot \sqrt[3]{y^2}}{z^4} \right)

We can simplify this logarithmic expression using logarithmic properties. Let’s break it down step by step:

Step 1: Simplify the radicals

x3=x32,andy23=y23\sqrt{x^3} = x^{\frac{3}{2}}, \quad \text{and} \quad \sqrt[3]{y^2} = y^{\frac{2}{3}}

Thus, the expression becomes: log(x32y23z4)\log \left( \frac{x^{\frac{3}{2}} \cdot y^{\frac{2}{3}}}{z^4} \right)

Step 2: Apply the logarithmic property for division

log(AB)=log(A)log(B)\log \left( \frac{A}{B} \right) = \log(A) - \log(B)

This gives: log(x32y23)log(z4)\log \left( x^{\frac{3}{2}} \cdot y^{\frac{2}{3}} \right) - \log(z^4)

Step 3: Apply the logarithmic property for multiplication

log(AB)=log(A)+log(B)\log(AB) = \log(A) + \log(B)

This gives: log(x32)+log(y23)log(z4)\log(x^{\frac{3}{2}}) + \log(y^{\frac{2}{3}}) - \log(z^4)

Step 4: Apply the logarithmic power rule

log(An)=nlog(A)\log(A^n) = n \log(A)

Thus: 32log(x)+23log(y)4log(z)\frac{3}{2} \log(x) + \frac{2}{3} \log(y) - 4 \log(z)

This is the simplified form of the given logarithmic expression.


Would you like further clarification on any step, or do you have other questions?

Related Questions:

  1. How can we apply these logarithmic properties to more complex expressions?
  2. What are the properties of logarithms that are most commonly used?
  3. How can the logarithmic power rule help in solving exponential equations?
  4. What happens if the logarithmic base is not specified in expressions?
  5. How does the change of base formula for logarithms work?

Tip:

When simplifying logarithmic expressions, always try to apply the power rule and product rule to break down complex terms into manageable parts.

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Math Problem Analysis

Mathematical Concepts

Algebra
Logarithmic Properties
Radicals

Formulas

log(A/B) = log(A) - log(B)
log(AB) = log(A) + log(B)
log(A^n) = n * log(A)
√(x³) = x^(3/2), ³√(y²) = y^(2/3)

Theorems

Logarithmic Rules

Suitable Grade Level

Grades 10-12