Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. \(\ln \left(\frac{a^{-2}}{b^{-4} c^{5}}\right)\)

Step-by-Step Solution

Answer

The expanded logarithm is \(-2\ln a + 4\ln b - 5\ln c\).

Step 1: Identify the Function Structure

The expression is a natural logarithm of a quotient: \(\ln \left(\frac{a^{-2}}{b^{-4} c^{5}}\right)\). This can be rewritten using the properties of logarithms as the difference between the natural logarithm of the numerator and the denominator.

Step 2: Apply the Quotient Rule

Using the quotient property of logarithms, \(\ln \left(\frac{M}{N}\right) = \ln M - \ln N\), rewrite the expression: \(\ln(a^{-2}) - \ln(b^{-4}c^{5})\).

Step 3: Apply the Product Rule

The denominator contains a product of terms, so use the product property of logarithms: \(\ln(MN) = \ln M + \ln N\). This rewrites the second term as \(\ln(b^{-4}c^{5}) = \ln(b^{-4}) + \ln(c^{5})\).

Step 4: Apply the Power Rule

Each logarithm with an exponent can be expanded using the power property: \(\ln(M^p) = p\ln M\). Thus, we have: \(-2\ln a - \left(-4\ln b + 5\ln c\right)\).

Step 5: Distribute the Negative Sign

Distribute the negative sign in the expression: \(-2\ln a + 4\ln b - 5\ln c\). This completes the expansion.

Key Concepts

Product Rule

The product rule is a key property when working with logarithms. It helps us break down complex logarithmic expressions involving a product inside the logarithm. This rule states that the logarithm of a product is equal to the sum of the logarithms of the individual terms.

• If you have an expression \(\ln (MN)\), it is the same as \(\ln M + \ln N\).

Consider the example in this exercise, where part of the expression is inherited from the denominator: \(\ln (b^{-4} c^{5})\). Applying the product rule, we break it down to: \(\ln (b^{-4}) + \ln (c^{5})\). This simplification allows us to handle each term separately and makes it easier to apply further rules.

Quotient Rule

In the world of logarithms, the quotient rule helps us manage a fraction inside a logarithmic expression. It converts a logarithm of a quotient into the difference of two logarithms.

• The rule states: \(\ln \left(\frac{M}{N}\right) = \ln M - \ln N\).

In the given exercise, the expression starts as a natural logarithm of a fraction: \(\ln \left(\frac{a^{-2}}{b^{-4} c^{5}}\right)\). By applying the quotient rule, the expression is split into: \(\ln (a^{-2}) - \ln (b^{-4} c^{5})\). This transformation simplifies the problem, allowing us to handle the numerator and denominator independently. The ensuing steps leverage this expansion for further simplification.

Power Rule

The power rule is a valuable tool when simplifying logarithmic expressions. This property allows us to "bring out" exponents in logarithms as coefficients, making expressions more manageable.

• This rule can be written as: \(\ln (M^p) = p \ln M\).

In our example, after applying the product and quotient rules, we have \(\ln (a^{-2})\), \(\ln (b^{-4})\) and \(\ln (c^5)\). By employing the power rule, each term gets expanded as follows:

• \(\ln (a^{-2}) = -2 \ln a\)
• \(\ln (b^{-4}) = -4 \ln b\)
• \(\ln (c^5) = 5 \ln c\)

These adjustments simplify computations by transforming exponential factors into multiplications, thus completing the expression's expansion.

Logarithmic Expansion

Logarithmic expansion involves breaking down a logarithmic expression into its simplest components using properties such as the product, quotient, and power rules. It aims to express the log of a complex expression as simple sums, differences, and multiples of individual logs.

• It streamlines complex log expressions into more digestible parts.

In this exercise, logarithmic expansion begins with the original expression: \(\ln \left(\frac{a^{-2}}{b^{-4} c^{5}}\right)\). By systematically applying the quotient, product, and power rules, you achieve \(-2 \ln a + 4 \ln b - 5 \ln c\). This final expression reflects the success of logarithmic expansion, simplifying the initial logarithmic fraction into straightforward log terms. Logarithmic expansion is particularly useful in calculus and algebra to simplify expressions before derivatives or integrations, ensuring smoother mathematical manipulations.