Use properties of logarithms to expand each logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator \( \log _{b}\left(x^{5} z\right) \) \( \log _{b}\left(x^{5} z\right)=\square \)

To expand the expression \( \log _{b}\left(x^{5} z\right) \), we can use the properties of logarithms. First, we apply the product property of logarithms, which states that \( \log_b(mn) = \log_b(m) + \log_b(n) \). This gives us: \[ \log _{b}\left(x^{5}\right) + \log _{b}\left(z\right) \] Next, we use the power property of logarithms, which states that \( \log_b(m^n) = n\log_b(m) \). Applying this to \( \log_b(x^5) \), we get: \[ 5 \log _{b}(x) + \log _{b}(z) \] So, the final expanded form is: \[ \log _{b}\left(x^{5} z\right) = 5 \log _{b}(x) + \log _{b}(z) \] For evaluation without a calculator, if we assign some arbitrary values, such as \( b = 10\), \( x = 2\), and \( z = 5\), we can evaluate it as follows: \[ \log_{10}(2^5) + \log_{10}(5) = 5\log_{10}(2) + \log_{10}(5) \approx 5(0.301) + 0.699 \approx 1.504 \]