Questions: Write the expression as a logarithm of a single expression. Simplify the final answer if possible. ln 3 x^8 - ln 3 x^4 ln 3 x^8 - ln 3 x^4 = ln (Simplify your answer.)

Solution Steps

To combine the given logarithmic expressions into a single logarithm, we can use the properties of logarithms. Specifically, the difference of two logarithms can be expressed as the logarithm of a quotient. Therefore, we can rewrite the expression \(\ln 3 x^{8} - \ln 3 x^{4}\) as \(\ln \left(\frac{3 x^{8}}{3 x^{4}}\right)\). Simplifying the fraction inside the logarithm will give us the final answer.

We start with the expression: \[ \ln(3 x^{8}) - \ln(3 x^{4}) \] Using the property of logarithms that states \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\), we can rewrite the expression as: \[ \ln\left(\frac{3 x^{8}}{3 x^{4}}\right) \]

Next, we simplify the fraction inside the logarithm: \[ \frac{3 x^{8}}{3 x^{4}} = \frac{x^{8}}{x^{4}} = x^{8-4} = x^{4} \] Thus, we have: \[ \ln\left(x^{4}\right) \]

Using the property of logarithms that states \(\ln(a^{b}) = b \ln(a)\), we can further simplify: \[ \ln\left(x^{4}\right) = 4 \ln(x) \]

Final Answer