Problem 62 - College Algebra

Question

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ 4 \ln x+7 \ln y-3 \ln z $$

Step-by-Step Solution

Answer

The expression condensed into a single logarithm with a coefficient of 1 is \( \ln (\frac{x^4 y^7}{z^3}) \)

Step 1: Applying the Power Rule

First apply the power rule, which states that \( a \ln b = \ln b^a \). Therefore, the expression becomes \( \ln x^4+\ln y^7-\ln z^3 \)

Step 2: Applying the Product Rule

Next, apply the product rule, which states that \( \ln a + \ln b = \ln ab \). Similarly, when subtracting, it means dividing. So, \( \ln a - \ln b = \ln(\frac{a}{b}) \). Therefore, the expression becomes \( \ln (\frac{x^4 y^7}{z^3}) \)

Step 3: Result

So, the condensed logarithmic expression is \( \ln (\frac{x^4 y^7}{z^3}) \)

Key Concepts

Power Rule of Logarithms

Understanding the power rule of logarithms is essential for working with logarithmic functions. This rule is based on the mathematical property that relates an exponent within a logarithm. The power rule states that multiplying a logarithm by a constant is equivalent to raising the argument of the logarithm to the power of that constant. In mathematical terms, the rule is expressed as:
\( a \log_b(x) = \log_b(x^a) \).

• It simplifies expressions with coefficients in front of the logarithm.
• It helps in rendering logarithmic equations more manageable.
• Applying it correctly can condense multiple logarithmic terms into a single term.

For example, consider the logarithmic expression \( 3 \log_2(5) \). According to the power rule, it can be rewritten as \( \log_2(5^3) \), thereby simplifying the expression.

Product Rule of Logarithms

The product rule of logarithms is another fundamental concept necessary for manipulating logarithmic expressions. This rule connects the operation of multiplication within the logarithmic argument to addition outside of it. It can be stated as:
\( \log_b(m) + \log_b(n) = \log_b(m \times n) \).

• It is used for combining two logarithms with the same base that are being added.
• Subtraction of logarithms is interpreted as division within the argument.
• This rule facilitates the condensation of expressions by turning products into sums.

Applying the product rule, if you have \( \log_2(3) + \log_2(4) \), it converts to a single logarithmic expression as \( \log_2(3 \times 4) \), simplifying the initial expression substantially.

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves combining multiple logarithmic terms into a single term. To achieve this, one must be adept at applying both the power and product rules of logarithms, as well as the quotient rule, which relates to the division of logarithmic expressions.

Quotient Rule in Condensation

Within condensing, the quotient rule comes in handy when logarithms are being subtracted, since it states: \( \log_b(m) - \log_b(n) = \log_b(\frac{m}{n}) \).
In practice, to condense an expression like \( \log(x) + 2\log(y) - 3\log(z) \), you would first apply the power rule to the coefficients, then combine the terms using the product and quotient rules to obtain:
\( \log(\frac{x \times y^2}{z^3}) \).

• The goal of condensation is to have a single logarithmic statement.
• Properly condensed forms are critical for solving logarithmic equations and comparisons.
• Understanding the relationship between the rules helps to condense expressions accurately.

This process is not just a simplification technique but also vital in higher-level mathematics, including calculus and beyond.