- Use the base-change formula \(\log_b a = \frac{\log_c a}{\log_c b}\) to simplify the given expression \(\frac{\ln 6}{\ln 3} = \log_3 6\).
- Analyze each option using logarithm identities and base-change properties.
- Conclude that \(\frac{\log 6}{\log 3}\), \(\log_3 6\), and \(\frac{\log_5 6}{\log_5 3}\) are equivalent to \(\frac{\ln 6}{\ln 3}\).
- The final answer consists of the expressions: \(\boxed{\frac{\log 6}{\log 3}, \log_3 6, \frac{\log_5 6}{\log_5 3}}\).
Explanation
- Understanding the Base-Change Formula
We aim to identify which of the given expressions are equivalent to the original expression \(\frac{\ln 6}{\ln 3}\). The expression is a quotient of two natural logarithms. Using the base-change formula for logarithms, which states \(\log_b a = \frac{\log_c a}{\log_c b}\), we can rewrite \(\frac{\ln 6}{\ln 3}\) as \(\log_3 6\). Understanding this fundamental identity will allow us to evaluate each provided option.
- Evaluating Given Expressions
We evaluate each expression given in the list to determine if it is equivalent to \(\log_3 6\).
- \(\frac{\log 6}{\log 3}\): Here, \(\log\) denotes the logarithm with base 10 (common logarithm). By the base-change formula, \(\frac{\log 6}{\log 3} = \log_3 6\). This is equivalent.
- \(\ln 2\): The original expression is \(\frac{\ln 6}{\ln 3} = \frac{\ln(2 \times 3)}{\ln 3} = \frac{\ln 2 + \ln 3}{\ln 3} = \frac{\ln 2}{\ln 3} + 1\). This is clearly not equal to \(\ln 2\).
- \(\log_3 6\): As derived from the definition of the base-change formula, this is the original expression. This is equivalent.
- \(\ln 6 - \ln 3\): By log properties, \(\ln 6 - \ln 3 = \ln(\frac{6}{3}) = \ln 2\). This is not equivalent.
- \(\log_6 3\): By the base-change formula, \(\log_6 3 = \frac{\ln 3}{\ln 6} = \frac{1}{\log_3 6}\). This is the reciprocal, so it is not equivalent.
- \(\frac{\log_5 6}{\log_5 3}\): By the base-change formula with base 5, \(\frac{\log_5 6}{\log_5 3} = \log_3 6\). This is equivalent.
- Conclusion
Based on the step-by-step evaluation, the equivalent expressions are \(\frac{\log 6}{\log 3}\), \(\log_3 6\), and \(\frac{\log_5 6}{\log_5 3}\). These all represent the value of \(\log_3 6\), which is equivalent to \(\frac{\ln 6}{\ln 3}\). Thus, the correct choices are those listed.
Examples
Logarithms are essential in finance for calculating compound interest and in science for measuring pH levels or decibels. For example, if you want to determine how many years it will take for an investment to grow by a certain factor, you might use logarithms to solve equations like \(3^x = 6\), which leads to \(x = \log_3 6 = \frac{\ln 6}{\ln 3}\). Mastering these transformations allows for simplified computations across various scientific and economic models.