use properties of logarithms to condense the logarithmic expression below. write the expression as a single…

use properties of logarithms to condense the logarithmic expression below. write the expression as a single logarithm whose coefficient is 1. where possible, evaluate logarithmic expressions.\n4 ln x + 2 ln y - 3 ln z\n\n4 ln x + 2 ln y - 3 ln z = \\square (simplify your answer.)

use properties of logarithms to condense the logarithmic expression below. write the expression as a single logarithm whose coefficient is 1. where possible, evaluate logarithmic expressions.\n4 ln x + 2 ln y - 3 ln z\n\n4 ln x + 2 ln y - 3 ln z = \\square (simplify your answer.)

Answer

Explanation:

Step1: Apply power rule of logarithms

The power rule of logarithms states that (a\ln b=\ln(b^a)). Applying this to each term:

  • For (4\ln x), we get (\ln(x^4))
  • For (2\ln y), we get (\ln(y^2))
  • For (- 3\ln z), we get (\ln(z^{-3})) or (\frac{1}{\ln(z^{3})}) (but we will use the addition/subtraction rules next)

So the expression becomes (\ln(x^4)+\ln(y^2)-\ln(z^3))

Step2: Apply product rule of logarithms

The product rule of logarithms states that (\ln a+\ln b = \ln(ab)). Applying this to (\ln(x^4)+\ln(y^2)), we get (\ln(x^4y^2))

Step3: Apply quotient rule of logarithms

The quotient rule of logarithms states that (\ln a-\ln b=\ln\left(\frac{a}{b}\right)). Applying this to (\ln(x^4y^2)-\ln(z^3)), we get (\ln\left(\frac{x^4y^2}{z^3}\right))

Answer:

(\ln\left(\frac{x^4y^2}{z^3}\right))