Question - Simplifying Logarithmic Expressions

For the given logarithmic expression:

\(\log 7 + \log 2 + \log \frac{1}{40} + \log \frac{1}{7}\)

We can simplify using the properties of logarithms:

Combine \(\log 7\) and \(\log \frac{1}{7}\) using \(\log a + \log \frac{1}{a} = \log 1 = 0\):

\(\log 7 + \log \frac{1}{7} = \log 1 = 0\)

Combine \(\log 2\) and \(\log \frac{1}{40}\) by rewriting \(\frac{1}{40}\) as \(2^{-3}\), then use the property \(\log a^b = b \log a\):

\(\log 2 + \log 2^{-3} = \log 2 + (-3)\log 2 = \log 2 - 3 \log 2\)

\(\log 2 - 3 \log 2 = -2 \log 2\)

Since \(\log 7 + \log \frac{1}{7} = 0\), all that's left is:

\(-2 \log 2\)