1. Express the following logarithms in terms of ln 2 and ln 3.
b. ln(4/9)

Verified step by step guidance

1. Recall the logarithm property that allows you to express the logarithm of a quotient as the difference of logarithms: \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\).
2. Apply this property to the given expression: \(\ln\left(\frac{4}{9}\right) = \ln 4 - \ln 9\).
3. Express 4 and 9 in terms of their prime factors: \(4 = 2^2\) and \(9 = 3^2\).
4. Use the logarithm power rule, which states \(\ln(a^b) = b \ln a\), to rewrite \(\ln 4\) and \(\ln 9\) as \(2 \ln 2\) and \(2 \ln 3\) respectively.
5. Combine these results to express the original logarithm entirely in terms of \(\ln 2\) and \(\ln 3\): \(\ln\left(\frac{4}{9}\right) = 2 \ln 2 - 2 \ln 3\).

Key Concepts

Properties of Logarithms

Logarithms have key properties such as the product, quotient, and power rules. The quotient rule states that ln(a/b) = ln(a) - ln(b), allowing the expression of logarithms of fractions as differences of logarithms.

Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e. It is commonly used in calculus and can be manipulated using its properties to simplify expressions involving constants like 2 and 3.

Expressing Numbers as Powers or Products

To rewrite logarithms in terms of ln 2 and ln 3, numbers like 4 and 9 should be expressed as powers or products of 2 and 3, e.g., 4 = 2^2 and 9 = 3^2, enabling the use of the power rule for logarithms.