Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \( \log _{5}(2 \cdot 13) \)

To expand the expression \( \log_{5}(2 \cdot 13) \), we can use the product property of logarithms, which states that \( \log_b(m \cdot n) = \log_b(m) + \log_b(n) \). Applying this property: \[ \log_{5}(2 \cdot 13) = \log_{5}(2) + \log_{5}(13) \] Unfortunately, we cannot simplify \( \log_{5}(2) \) or \( \log_{5}(13) \) into a numeric form without a calculator, so we leave these logarithms in their expanded form. Hence, your final expanded expression is \[ \log_{5}(2) + \log_{5}(13). \] Remember, if you're ever in a pinch and need to estimate values, knowing that \( \log_{5}(2) \) is less than 1 and \( \log_{5}(13) \) is greater than 1 can help you get a rough idea of the size of the result!