Question

To express the quantity \(\ln 5 + 5\ln 3\) as a single logarithm.

Short answer

The given logarithm function is \(\ln 5 + 5\ln 3\).

Step-by-step solution

Given data

The given logarithm function is \(\ln 5 + 5\ln 3\).

Concept of Law of logarithm

Laws of logarithm:

Quotient law:\({\log _a}\left( {\frac{x}{y}} \right) = {\log _a}x - {\log _a}y\)

Product law:\({\log _a}(xy) = {\log _a}x + {\log _a}y\)

Power law:\({\log _a}\left( {{x^r}} \right) = r{\log _a}x\), where\(r\)is any real number.

Calculation of the function \(\ln 5 + 5\ln 3\)

Use the law of logarithm, \({\ln _e}\left( {{x^r}} \right) = r\ln {n_e}x\), to express \(5\ln 3\) as \(\ln {3^5}\).

That is, \(5\ln 3 = \ln (243)\).

Therefore, the given quantity becomes \(\ln 5 + \ln (243)\).

By using the law of logarithm, \({\ln _e}(xy) = {\ln _e}x + {\ln _e}y\), it can be written that \(\ln 5 + \ln (243) = \ln (5 \times 243)\).

Thus, the quantity \(\ln 5 + \ln 243\) can be expressed as \(\underline {\ln 1215} \).