2. Express the following logarithms in terms of ln 5 and ln 7.
b. ln 9.8

Verified step by step guidance

1. Start by expressing 9.8 as a product or quotient of numbers involving 5 and 7. Notice that 9.8 can be written as \( 9.8 = \frac{49}{5} \) because \( 49 = 7^2 \) and \( 9.8 = 9.8 \).
2. Rewrite \( \ln 9.8 \) using the expression found: \( \ln 9.8 = \ln \left( \frac{49}{5} \right) \).
3. Use the logarithm property for division: \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \). So, \( \ln 9.8 = \ln 49 - \ln 5 \).
4. Express \( \ln 49 \) in terms of \( \ln 7 \) using the power rule for logarithms: \( \ln 49 = \ln (7^2) = 2 \ln 7 \).
5. Combine all parts to write \( \ln 9.8 \) fully in terms of \( \ln 5 \) and \( \ln 7 \): \( \ln 9.8 = 2 \ln 7 - \ln 5 \).

Key Concepts

Properties of Logarithms

Logarithmic properties such as the product, quotient, and power rules allow us to rewrite logarithms of complex numbers in terms of simpler components. For example, ln(ab) = ln a + ln b and ln(a^n) = n ln a. These properties are essential for expressing ln 9.8 in terms of ln 5 and ln 7.

Prime Factorization and Approximation

Breaking down numbers into products of prime factors or close approximations helps in rewriting logarithms. Since 9.8 is not a simple product of 5 and 7, approximating or expressing 9.8 as a product or quotient involving 5 and 7 is necessary to use known logarithms effectively.

Natural Logarithm (ln) Definition

The natural logarithm ln x is the logarithm to the base e, where e is Euler's number (~2.718). Understanding ln as the inverse of the exponential function helps in manipulating and combining logarithmic expressions, especially when expressing one logarithm in terms of others.