Answer each of the following. Write log₃ 12 in terms of natural logarithms using the change-of-base theorem.

Verified step by step guidance

1. Recall the change-of-base theorem, which states that for any positive numbers \(a\), \(b\), and base \(c\) (where \(a \neq 1\) and \(c \neq 1\)), the logarithm \(\log_a b\) can be rewritten as \(\frac{\log_c b}{\log_c a}\).
2. In this problem, we want to express \(\log_3 12\) in terms of natural logarithms, which means using the natural logarithm function \(\ln\) as the new base.
3. Apply the change-of-base formula with \(a = 3\), \(b = 12\), and \(c = e\) (the base of natural logarithms), so \(\log_3 12 = \frac{\ln 12}{\ln 3}\).
4. Write the expression clearly as \(\log_3 12 = \frac{\ln 12}{\ln 3}\), which is the logarithm base 3 of 12 expressed in terms of natural logarithms.
5. This expression can now be used for further calculations or evaluations using a calculator that has the natural logarithm function.

Key Concepts

Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) is the exponent x such that b^x = a. Understanding this definition is essential for manipulating and converting logarithmic expressions.

Change-of-Base Theorem

The change-of-base theorem allows you to rewrite a logarithm with any base in terms of logarithms with a different base. It states that log_b(a) = log_c(a) / log_c(b), where c is a new base, often chosen as e (natural logarithm) or 10 for convenience.

Natural Logarithms

Natural logarithms use the base e (approximately 2.718) and are denoted as ln(x). They are widely used in calculus and higher mathematics. Expressing logarithms in terms of natural logs simplifies calculations and connects logarithmic functions to exponential growth and decay.