Condense the logarithm. You must show thorough, handwritten work. You may only use scratch paper, a writing utensil, and a calculator that does not have a computer algebra system. \( \log x-5 \log y \)

To condense the expression \( \log x - 5 \log y \), we can use the properties of logarithms. The first property states that \( a \log b = \log b^a \). Applying this property: 1. Take the \( -5 \log y \) and rewrite it: \[ -5 \log y = \log y^{-5} \] 2. Substitute this back into the original expression: \[ \log x - 5 \log y = \log x + \log y^{-5} \] 3. Now, we can apply the property that \( \log a + \log b = \log(ab) \): \[ \log x + \log y^{-5} = \log(xy^{-5}) \] So, the condensed form of \( \log x - 5 \log y \) is: \[ \log(xy^{-5}) \] --- Did you know that logarithms were first introduced in the early 17th century by John Napier? His invention of logarithms transformed the fields of mathematics and engineering by making complex calculations much easier, especially before the invention of calculators! If you’re wondering where logarithms fit into the real world, think about their applications! Logarithms are crucial in fields like acoustics for measuring sound intensity (decibels), in biology for measuring population growth rates, and in finance for calculating compound interest. Their relevance extends to many other scientific and engineering disciplines!