Questions: 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. 4 ln x + 3 ln y - 2 ln z 4 ln x + 3 ln y - 2 ln z =

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.

4 ln x + 3 ln y - 2 ln z
4 ln x + 3 ln y - 2 ln z =

Transcript text: 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. \[ \begin{array}{r} 4 \ln x+3 \ln y-2 \ln z \\ 4 \ln x+3 \ln y-2 \ln z= \end{array} \] $\square$

Solution

Solution Steps

To condense the given logarithmic expression into a single logarithm, we can use the properties of logarithms. Specifically, we will use the power rule, which states that $a \ln b = \ln b^a$, and the product and quotient rules, which state that $\ln a + \ln b = \ln (ab)$ and $\ln a - \ln b = \ln (a/b)$, respectively.

Solution Approach

Apply the power rule to each term.
Combine the terms using the product and quotient rules.

Step 1: Apply the Power Rule

We start with the expression $4 \ln x + 3 \ln y - 2 \ln z$. Using the power rule of logarithms, we can rewrite each term: \[ 4 \ln x = \ln(x^4), \quad 3 \ln y = \ln(y^3), \quad -2 \ln z = \ln(z^{-2}) \] Thus, the expression becomes: \[ \ln(x^4) + \ln(y^3) + \ln(z^{-2}) \]

Step 2: Combine Using Product and Quotient Rules

Next, we combine the logarithmic terms using the product and quotient rules. The product rule states that $\ln a + \ln b = \ln(ab)$ and the quotient rule states that $\ln a - \ln b = \ln\left(\frac{a}{b}\right)$. Therefore, we can combine the terms as follows: \[ \ln(x^4) + \ln(y^3) - \ln(z^2) = \ln\left(\frac{x^4 y^3}{z^2}\right) \]