Find each value. If applicable, give an approximation to four decimal places. ln 84 - ln 17

Verified step by step guidance

1. Recall the logarithmic property that states the difference of two natural logarithms can be expressed as the logarithm of a quotient: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\).
2. Apply this property to the given expression: \(\ln 84 - \ln 17 = \ln \left( \frac{84}{17} \right)\).
3. Calculate the quotient inside the logarithm: \(\frac{84}{17}\).
4. Evaluate the natural logarithm of the quotient: \(\ln \left( \frac{84}{17} \right)\).
5. If needed, use a calculator to approximate the value of \(\ln \left( \frac{84}{17} \right)\) to four decimal places.

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the difference rule: ln(a) - ln(b) = ln(a/b). This allows combining or breaking down logarithmic expressions to make calculations easier.

Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e (approximately 2.718). It is the inverse function of the exponential function e^x and is commonly used in calculus and algebra.

Approximation of Logarithmic Values

When exact values are not easily found, logarithmic expressions can be approximated using calculators or tables. Approximations are often rounded to a specified number of decimal places for clarity and precision.