To simplify the expression \( \ln 7 + 3 \ln w - 2 \ln x \), we will use the properties of logarithms:
- Power Rule of Logarithms: \( n \ln a = \ln(a^n) \)
- Product Rule of Logarithms: \( \ln a + \ln b = \ln(ab) \)
- Quotient Rule of Logarithms: \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \)
Let's apply these rules step by step:
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First, we can apply the power rule to \( 3 \ln w \) and \( -2 \ln x \): \[ 3 \ln w = \ln(w^3) \quad \text{and} \quad -2 \ln x = \ln\left(\frac{1}{x^2}\right) \]
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Now, we can rewrite the overall expression: \[ \ln 7 + 3 \ln w - 2 \ln x = \ln 7 + \ln(w^3) + \ln\left(\frac{1}{x^2}\right) \]
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Next, we simplify it using the product and quotient rules: \[ \ln 7 + \ln(w^3) = \ln(7w^3) \]
Therefore, we have: \[ \ln(7w^3) + \ln\left(\frac{1}{x^2}\right) = \ln(7w^3) - \ln(x^2) = \ln\left(\frac{7w^3}{x^2}\right) \]
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The final result is: \[ \ln\left(\frac{7w^3}{x^2}\right) \]
Now let's evaluate the options:
- Option #1: \(\ln(21w^2x)\) – This does not match our result.
- Option #2: \(\ln(7w^3x^2)\) – This does not match our result.
- Option #3: \(\ln(73wx)\) – This does not match our result.
None of the options provided are correct based on our simplified expression. The simplified expression is \(\ln\left(\frac{7w^3}{x^2}\right)\).