Question

Use the properties of logarithms to approximate the indicated logarithms, given that \(\ln 2 \approx 0.6931\) and \(\ln 3 \approx 1.0986 .\) (a) \(\ln 6\) (b) \(\ln \frac{2}{3}\) (c) \(\ln 81\) (d) \(\ln \sqrt{3}\)

Short answer

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

Step-by-step solution

Compute \(\ln 6\)

Use the product rule for logarithms, which states that \( \ln (a \cdot b) = \ln a + \ln b \). Thus, we can rewrite \(\ln 6\) as \(\ln (2 \cdot 3) = \ln 2 + \ln 3\), which equals 0.6931 + 1.0986 = 1.7917.

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

Use the quotient rule for logarithms, which states that \(\ln \frac{a}{b} = \ln a - \ln b\). So, \(\ln \frac{2}{3}\) can be rewritten as \(\ln 2 - \ln 3 = 0.6931 - 1.0986 = -0.4055.\)

Compute \(\ln 81\)

The number 81 can be expressed as \(3^4\). Use the power rule, which states that \(\ln a^n = n \cdot \ln a\). Thus, \(\ln 81 = \ln 3^4 = 4 \cdot \ln 3 = 4 \cdot 1.0986 = 4.3944.\

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

\(\sqrt{3}\) can be written as \(3^{1/2}\). Using the power rule again, we can rewrite \(\ln \sqrt{3} = \ln 3^{1/2} = \frac{1}{2} \cdot \ln 3 = 0.5 \cdot 1.0986 = 0.5493.\)

Key concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln x\), is a mathematical function that represents the logarithm to the base \(e\), where \(e\) is an irrational and transcendental constant approximately equal to 2.71828. In simple terms, when you take the natural logarithm of a number, you’re asking the question: 'To what power do I need to raise \(e\) to get this number?'.

For example, if you have \(\ln 6\), you're essentially looking for the power that \(e\) must be raised to in order to produce 6. The natural logarithm has a direct relationship with exponential functions, and it has unique properties that make it particularly useful in calculus and solving problems involving growth and decay.

Logarithm Product Rule

One of the fundamental properties of logarithms is the logarithm product rule. The rule states that the log of a product is equal to the sum of the logs of its factors. Formally, \(\ln(ab) = \ln a + \ln b\).

Whenever we encounter the natural logarithm of a multiplication, such as \(\ln (2 \cdot 3)\), this rule permits us to simplify the expression by splitting it into separate logs: \(\ln 2 + \ln 3\). This is particularly useful when approximating logarithms since we can break down complicated expressions into sums of logarithms with known values.

Logarithm Quotient Rule

The logarithm quotient rule is another essential property that simplifies the natural log of a division into a difference of two logs. The rule is formulated as \(\ln\frac{a}{b} = \ln a - \ln b\).

In practice, if we want to calculate \(\ln\frac{2}{3}\), we can use this rule to transform it into \(\ln 2 - \ln 3\). This can then be evaluated using known logarithm values, making it easier to handle expressions where direct calculation might be difficult.

Logarithm Power Rule

The logarithm power rule is a powerful tool that links exponents and logarithms. This rule states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number itself: \(\ln(a^n) = n\ln a\).

Using this rule, we can simplify expressions such as \(\ln 81\), knowing that 81 is \(3^4\). Applying the power rule gives us \(\ln(3^4) = 4\cdot\ln 3\). Similarly, the rule helps us to find \(\ln\sqrt{3}\), which is \(\ln(3^{1/2}) = \frac{1}{2}\cdot\ln 3\). These properties allow us to break down complex logarithmic expressions into simpler, more manageable components.