Question

rewrite the following logarithmic expression as one logarithm. 12 ln(x) + ln(y) + 3 ln(z)

Explanation:

Step1: Apply power rule of logarithms

The power rule states that \( n\ln(a)=\ln(a^n) \). So, for \( 12\ln(x) \), we get \( \ln(x^{12}) \), and for \( 3\ln(z) \), we get \( \ln(z^3) \). The expression now becomes \( \ln(x^{12})+\ln(y)+\ln(z^3) \).

Step2: Apply product rule of logarithms

The product rule states that \( \ln(a)+\ln(b)=\ln(ab) \). First, combine \( \ln(x^{12}) \) and \( \ln(y) \): \( \ln(x^{12}y) \). Then combine this with \( \ln(z^3) \): \( \ln(x^{12}y z^3) \).

Answer:

\( \ln(x^{12}yz^3) \)