Use the approximations ln2 ≈0.693 and ln3 ≈1.099 together with the properties stated in Theorem

step 1 For this exercise, we need to use these rules. Part A, log 6 equals log 2 times 3, which equals

Use the approximations ln2 ≈0.693 and ln3 ≈1.099 together...

Use the approximations $\ln 2 \approx 0.693$ and $\ln 3 \approx 1.099$ together with the properties stated in Theorem A to calculate approximations to each of the following. For example, $\ln 6=$ $\ln (2 \cdot 3)=\ln 2+\ln 3 \approx 0.693+1.099=1.792$. (a) $\ln 6$ (b) $\ln 1.5$ (c) In 81 (d) $\ln \sqrt{2}$ (e) $\ln \left(\frac{1}{36}\right)$ (f) $\ln 48$

Use the approximations $\ln 2 \approx 0.693$ and $\ln 3 \approx 1.099$ together with the properties stated in Theorem A to calculate approximations to each of the following. For example, $\ln 6=$ $\ln (2 \cdot 3)=\ln 2+\ln 3 \approx 0.693+1.099=1.792$.
(a) $\ln 6$
(b) $\ln 1.5$
(c) In 81
(d) $\ln \sqrt{2}$
(e) $\ln \left(\frac{1}{36}\right)$
(f) $\ln 48$

Key Concepts

Approximation Techniques

Approximation techniques involve substituting given numerical approximations of logarithms into the simplified expressions obtained from applying logarithmic properties. This method allows for practical numerical estimations when exact values are difficult or unnecessary to compute.

Power Rule

The power rule for logarithms shows that the logarithm of an exponentiated number can be expressed as the exponent times the logarithm of the base: ln(a^b) = b · ln a. This property is crucial for simplifying logarithmic expressions that involve radicals or exponentiation by rewriting them in a form that is easier to calculate.

Natural Logarithm

The natural logarithm, denoted by ln, is a logarithm with base e (approximately 2.71828). It is widely used in calculus, exponential growth/decay problems, and many real-world applications due to its natural properties and ease of differentiation and integration.

Product Rule

The product rule for logarithms states that the logarithm of a product is the sum of the logarithms of the factors, i.e., ln(ab) = ln a + ln b. This property is essential for breaking down the logarithms of composite numbers into simpler parts, often utilizing pre-known logarithmic values.

Quotient Rule

The quotient rule for logarithms indicates that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator, expressed as ln(a/b) = ln a - ln b. This rule is useful for simplifying expressions and solving equations where division of quantities is involved.

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