2. Express the following logarithms in terms of ln 5 and ln 7.
d. ln 1225

Verified step by step guidance

1. Start by factoring the number inside the logarithm to express it in terms of its prime factors or known bases. For 1225, find its prime factorization.
2. Recognize that 1225 can be written as a product of powers of 5 and 7, since 1225 = 35^2 and 35 = 5 \(\times\) 7.
3. Rewrite the logarithm using the property of logarithms that states \( \ln(a^b) = b \ln(a) \). So, \( \ln(1225) = \ln(35^2) = 2 \ln(35) \).
4. Next, use the logarithm product rule \( \ln(ab) = \ln(a) + \ln(b) \) to express \( \ln(35) \) as \( \ln(5) + \ln(7) \).
5. Combine all the steps to express \( \ln(1225) \) entirely in terms of \( \ln(5) \) and \( \ln(7) \) as \( 2 (\ln(5) + \ln(7)) \).

Key Concepts

Properties of Logarithms

Logarithmic properties such as the product, quotient, and power rules allow us to break down complex logarithms into simpler parts. For example, ln(ab) = ln a + ln b and ln(a^n) = n ln a. These rules help express logarithms in terms of known values.

Prime Factorization

Prime factorization involves expressing a number as a product of prime numbers. This is useful in logarithms to rewrite the argument as a product of primes or their powers, enabling the use of logarithmic properties to simplify expressions.

Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e. Understanding that ln is a specific logarithm function helps in manipulating and expressing logarithms in terms of known ln values, such as ln 5 and ln 7.