Math Problem Statement

Find a and b such that ln((x^2 * sqrt(x^3 * y^5)) / y^3) = a ln(x) + b ln(y).

Solution

It seems like you're trying to solve an equation involving logarithms. From the image, the problem asks to find the values of aa and bb such that:

ln(x2x3y5y3)=aln(x)+bln(y)\ln \left( \frac{x^2 \sqrt{x^3 y^5}}{y^3} \right) = a \ln(x) + b \ln(y)

To solve this, we will need to simplify the logarithmic expression on the left-hand side and compare the results with the form on the right-hand side.

Step-by-step process:

  1. Simplify the argument inside the logarithm: x2x3y5y3\frac{x^2 \sqrt{x^3 y^5}}{y^3} First, express the square root as a fractional exponent: x3y5=(x3y5)1/2=x3/2y5/2\sqrt{x^3 y^5} = (x^3 y^5)^{1/2} = x^{3/2} y^{5/2} So the entire expression becomes: x2x3/2y5/2y3=x2+3/2y5/23=x7/2y1/2\frac{x^2 x^{3/2} y^{5/2}}{y^3} = x^{2 + 3/2} y^{5/2 - 3} = x^{7/2} y^{-1/2}

  2. Apply the logarithm: Using the properties of logarithms: ln(x7/2y1/2)=ln(x7/2)+ln(y1/2)\ln \left( x^{7/2} y^{-1/2} \right) = \ln \left( x^{7/2} \right) + \ln \left( y^{-1/2} \right) Using the power rule ln(xk)=kln(x)\ln(x^k) = k \ln(x), this becomes: 72ln(x)12ln(y)\frac{7}{2} \ln(x) - \frac{1}{2} \ln(y)

  3. Compare this with the right-hand side: You are given that: ln(x2x3y5y3)=aln(x)+bln(y)\ln \left( \frac{x^2 \sqrt{x^3 y^5}}{y^3} \right) = a \ln(x) + b \ln(y) From the simplification above, we can now equate the coefficients of ln(x)\ln(x) and ln(y)\ln(y): a=72,b=12a = \frac{7}{2}, \quad b = -\frac{1}{2}

Thus, the values are: a=72,b=12a = \frac{7}{2}, \quad b = -\frac{1}{2}

Would you like further details on any of these steps? Let me know if you need help with similar problems!

Related questions:

  1. How do you apply the power rule for logarithms?
  2. What are the properties of logarithms used in this problem?
  3. How can you simplify an expression with multiple variables under a logarithm?
  4. What is the difference between the natural logarithm and other logarithmic bases?
  5. Can this method be applied to more complex logarithmic expressions?

Tip: Always remember that simplifying the argument of a logarithmic expression before applying the properties can make the problem much easier to solve!

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Math Problem Analysis

Mathematical Concepts

Logarithmic Simplification
Exponents and Roots

Formulas

ln(x^k) = k ln(x)
sqrt(a) = a^(1/2)
ln(a / b) = ln(a) - ln(b)
ln(ab) = ln(a) + ln(b)

Theorems

Properties of Logarithms

Suitable Grade Level

Grades 10-12