Question

In Exercises, use the properties of logarithms and the fact that \(\ln 2 \approx 0.6931\) and \(\ln 3 \approx 1.0986\) to approximate the logarithm. Then use a calculator to confirm your approximation. (a) \(\ln 6\) (b) \(\ln \frac{3}{2}\) (c) \(\ln 81\) (d) \(\ln \sqrt{3}\)

Short answer

\(\ln 6 \approx 1.7917, \ln \frac{3}{2} \approx 0.4055, \ln 81 \approx 4.3944, \ln \sqrt{3} \approx 0.5493\)

Step-by-step solution

Approximating \(\ln 6\)

From the product rule of logarithms (\(\ln ab = \ln a + \ln b\)), we know that \(\ln 6 = \ln(2*3) = \ln 2 + \ln 3 = 0.6931 + 1.0986 = 1.7917\)

Approximating \(\ln \frac{3}{2}\)

From the quotient rule of logarithms (\(\ln \frac{a}{b} = \ln a - \ln b\)), we get that \(\ln \frac{3}{2} = \ln 3 - \ln 2 = 1.0986 - 0.6931 = 0.4055\)

Approximating \(\ln 81\)

Rewriting \(81 = 3^4\), by the power rule of logarithms (\(\ln a^k = k \ln a\)), we get that \(\ln 81 = \ln (3^4) = 4 \ln 3 = 4 * 1.0986 = 4.3944\)

Approximating \(\ln \sqrt{3}\)

Rewriting \(\sqrt{3} = 3^{\frac{1}{2}}\), by the power rule of the logarithm, we get that \(\ln \sqrt{3} = \ln (3^{\frac{1}{2}}) = \frac{1}{2} \ln 3 = \frac{1}{2} * 1.0986 = 0.5493\)

Key concepts

Understanding the Product Rule in Logarithms

The product rule is a fundamental property of logarithms that can simplify the process of working with logarithms that involve multiplication. In essence, the product rule states that the logarithm of the product of two numbers is equal to the sum of the logarithms of those numbers. Mathematically, this is expressed as \( \ln(ab) = \ln a + \ln b \).

For example, when you have \( \ln 6 \), you can express 6 as the product of 2 and 3: \( 6 = 2 \times 3 \). Using the product rule, \( \ln 6 \) becomes \( \ln(2 \times 3) = \ln 2 + \ln 3 \). This separation into two simpler logarithms makes it easier to compute each logarithm, especially when you know the approximate values: \( \ln 2 \approx 0.6931 \) and \( \ln 3 \approx 1.0986 \). Thus, \( \ln 6 = 0.6931 + 1.0986 = 1.7917 \).

By understanding and applying the product rule, you can effectively break down more complex problems into simpler, additive components.

Mastering the Quotient Rule with Logarithms

The quotient rule is another essential property in logarithms that helps simplify division problems involving logarithms. This rule tells us that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. It is represented as \( \ln \left(\frac{a}{b}\right) = \ln a - \ln b \).

Consider the example \( \ln \frac{3}{2} \). Using the quotient rule, this becomes \( \ln 3 - \ln 2 \). By substituting the known values, \( \ln 3 \approx 1.0986 \) and \( \ln 2 \approx 0.6931 \), we calculate \( \ln \frac{3}{2} = 1.0986 - 0.6931 = 0.4055 \).

The simplicity of the quotient rule allows for the easy handling of logarithmic expressions that involve division, breaking them into straightforward, subtractive operations.

Exploring the Power Rule in Logarithms

The power rule is a critical logarithmic property used to manage expressions where a number is raised to a power. This rule states that the logarithm of a number raised to a power is the power times the logarithm of the number. It is expressed as \( \ln a^k = k \ln a \).

For instance, when calculating \( \ln 81 \), it's helpful to recognize that 81 can be written as \( 3^4 \). Applying the power rule, \( \ln 81 = 4 \ln 3 \). Using the given value for \( \ln 3 \), which is \( 1.0986 \), we calculate \( \ln 81 = 4 \times 1.0986 = 4.3944 \).

The power rule is also useful for roots, as shown in the example \( \ln \sqrt{3} \). The square root of a number can be expressed as raising that number to the half power \( (\frac{1}{2}) \). Thus, \( \ln \sqrt{3} = \ln (3^{\frac{1}{2}}) = \frac{1}{2} \ln 3 \). With \( \ln 3 \approx 1.0986 \), we find \( \ln \sqrt{3} = \frac{1}{2} \times 1.0986 = 0.5493 \).

Understanding the power rule allows for flexibility and efficiency when working with logarithmic problems, especially those involving powers and roots.