2. Express the following logarithms in terms of ln 5 and ln 7.
f. (ln35 + ln(1/7))/(ln25)

Verified step by step guidance

1. Recall the logarithm properties: \(\ln(ab) = \ln a + \ln b\) and \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\).
2. Rewrite \(\ln 35\) as \(\ln(5 \times 7)\), which becomes \(\ln 5 + \ln 7\) using the product rule.
3. Rewrite \(\ln \left(\frac{1}{7}\right)\) as \(\ln 1 - \ln 7\). Since \(\ln 1 = 0\), this simplifies to \(-\ln 7\).
4. Add the terms in the numerator: \((\ln 35 + \ln(1/7)) = (\ln 5 + \ln 7) + (-\ln 7) = \ln 5\).
5. Rewrite the denominator \(\ln 25\) as \(\ln(5^2) = 2 \ln 5\). Then express the entire fraction as \(\frac{\ln 5}{2 \ln 5}\).

Key Concepts

Properties of Logarithms

Logarithms have key properties such as the product rule (ln a + ln b = ln(ab)), the quotient rule (ln a - ln b = ln(a/b)), and the power rule (ln(a^b) = b ln a). These allow complex logarithmic expressions to be simplified or rewritten in terms of simpler logarithms.

Change of Base and Logarithm Simplification

Simplifying logarithmic expressions often involves rewriting numbers as products or powers of known bases. For example, expressing 35 as 5 × 7 or 25 as 5² helps rewrite logarithms in terms of ln 5 and ln 7, facilitating easier manipulation and evaluation.

Natural Logarithm (ln) and Its Use

The natural logarithm, denoted ln, is the logarithm with base e. It is commonly used in calculus and higher mathematics. Understanding how to express other logarithms in terms of ln 5 and ln 7 requires familiarity with ln’s properties and how to decompose numbers into factors involving 5 and 7.