Questions: Simplify the following expression. Write as a single logarithm or monomial. 5 ln z - 1/4 ln y + 7 ln x 5 ln z - 1/4 ln y + 7 ln x =

Solution Steps

To simplify the given expression, we can use the properties of logarithms. Specifically, we will use the power rule \(\log_b(a^c) = c \log_b(a)\) to move the coefficients inside the logarithms as exponents. Then, we will use the product rule \(\log_b(a) + \log_b(c) = \log_b(ac)\) and the quotient rule \(\log_b(a) - \log_b(c) = \log_b(a/c)\) to combine the logarithms into a single logarithm.

We start with the expression: \[ 5 \ln z - \frac{1}{4} \ln y + 7 \ln x \] Using the power rule of logarithms, \(\log_b(a^c) = c \log_b(a)\), we can rewrite the expression as: \[ \ln(z^5) - \ln(y^{1/4}) + \ln(x^7) \]

Next, we use the product rule \(\log_b(a) + \log_b(c) = \log_b(ac)\) and the quotient rule \(\log_b(a) - \log_b(c) = \log_b(a/c)\) to combine the logarithms into a single logarithm: \[ \ln(z^5) + \ln(x^7) - \ln(y^{1/4}) \] This can be further simplified to: \[ \ln\left(\frac{z^5 x^7}{y^{1/4}}\right) \]

Final Answer