Evaluate or simplify the expression without using a calculator. 4) \( 10 \log 7 \)

To simplify the expression \( 10 \log 7 \), we can utilize the properties of logarithms. The expression can be rewritten as \( \log(7^{10}) \) using the power rule of logarithms. This gives us \( 10 \log 7 = \log(7^{10}) \), which is the logarithm of \( 7^{10} \) or \( 282475249 \). Thus, \( 10 \log 7 = \log(282475249) \). For a fun fact, logs are incredibly useful in everyday applications, like in measuring sound intensity in decibels or the Richter scale for earthquakes! They help us understand vast ranges of values by compressing them into a more manageable scale. So, next time you hear about decibels or earthquakes, you know logs are behind the scenes doing the math magic! Another tip for dealing with logarithms is to remember that \( \log(1) = 0 \) and \( \log(10) = 1 \). These two values can help you quickly evaluate expressions involving logarithms without turning to a calculator. So keep these key points in mind as you simplify or evaluate logarithmic expressions!