Find each value. If applicable, give an approximation to four decimal places. ln 98 - ln 13

검증된 단계별 안내

Recall the logarithmic property that states the difference of two natural logarithms can be expressed as the logarithm of a quotient: \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\).

Apply this property to the given expression: \(\ln 98 - \ln 13 = \ln \left( \frac{98}{13} \right)\).

Calculate the quotient inside the logarithm: \(\frac{98}{13}\) (you can leave it as a fraction or convert it to a decimal for approximation).

Evaluate the natural logarithm of the quotient using a calculator or logarithm table: \(\ln \left( \frac{98}{13} \right)\).

If an approximation is required, round the result to four decimal places.

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주요 개념

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Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the difference of logarithms: ln(a) - ln(b) = ln(a/b). This property 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 for growth and decay problems.

Approximation of Logarithmic Values

When exact values of logarithms are not easily found, approximations using calculators or tables are used. Approximations are often rounded to a specified number of decimal places, such as four, to provide a practical and precise answer.