Questions: Write the expression as a sum, difference, or product of simpler logarithms. log6(5k) = □

Solution Steps

To express \(\log_{6}(5k)\) as a sum, difference, or product of simpler logarithms, we can use the properties of logarithms. Specifically, we can use the product rule of logarithms, which states that \(\log_b(xy) = \log_b(x) + \log_b(y)\).

To express \(\log_{6}(5k)\) as a sum of simpler logarithms, we use the product rule of logarithms: \[ \log_{b}(xy) = \log_{b}(x) + \log_{b}(y) \] Here, \(x = 5\) and \(y = k\), so: \[ \log_{6}(5k) = \log_{6}(5) + \log_{6}(k) \]

Using the change of base formula for logarithms, we can express \(\log_{6}(5)\) and \(\log_{6}(k)\) in terms of natural logarithms: \[ \log_{6}(5) = \frac{\log(5)}{\log(6)} \] \[ \log_{6}(k) = \frac{\log(k)}{\log(6)} \] Thus, the expression becomes: \[ \log_{6}(5k) = \frac{\log(5)}{\log(6)} + \frac{\log(k)}{\log(6)} \]

Final Answer