Focuses on the Quotient Rule, which states that the logarithm of a quotient is the difference of the logarithms. Includes examples of both expanding and condensing expressions.
For any valid base \(b\), \(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\). The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.
Use the Quotient Rule to expand the expression \(\log_7(\frac{x}{2})\).
$$\log_7(x) - \log_7(2)$$
Use the Quotient Rule to expand the expression \(\ln(\frac{e^2}{5})\).
The expansion is \(\ln(e^2) - \ln(5)\). Since \(\ln(e^2) = 2\), the expression simplifies to \(2 - \ln(5)\).
Condense the expression \(\log(30) - \log(3)\) into a single logarithm.
$$\log(\frac{30}{3}) = \log(10) = 1$$
Condense the expression \(\log_b(y) - \log_b(z)\) into a single logarithm.
$$\log_b(\frac{y}{z})$$
True or False: \(\frac{\log_b(M)}{\log_b(N)} = \log_b(M) - \log_b(N)\).
False. This is a common mistake. The Quotient Rule applies to the logarithm of a quotient, \(\log_b(\frac{M}{N})\), not a quotient of logarithms.
If \(\log_3(2) \approx 0.63\) and \(\log_3(10) \approx 2.10\), what is \(\log_3(5)\)?
