Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to condense the expression.$$\ln x-3 \ln (x+1)$$.

Step-by-Step Solution

Answer

The condensed form of the expression \(\ln x-3 \ln(x+1)\) using logarithm properties results in \(\ln\left(\frac{x}{(x+1)^3}\right)\).

Step 1: Applying the power rule

Identify instances where the power rule can be applied. In this case, the power rule can be applied to the second term in the expression. Changing the expression \(3 \ln(x+1)\) by applying the rule gives \(\ln{(x+1)^3}\). The expression now reads: \(\ln(x) - \ln{(x+1)^3}\)

Step 2: Applying the quotient rule

Recognize any chances to apply the quotient rule. This rule can be applied when there’s a subtraction between two logarithms. Apply the quotient rule to the current expression \(\ln(x) - \ln{(x+1)^3}\) to obtain \(\ln\left(\frac{x}{(x+1)^3}\right)\). This is the final, condensed form of the provided logarithmic expression.

Key Concepts

Properties of Logarithms

Logarithms are incredibly useful in simplifying complex expressions and solving exponential equations. The properties of logarithms are essential tools for these tasks.

These properties include:

• Product Rule: \(\log_b (MN) = \log_b M + \log_b N\).
• Quotient Rule: \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\).
• Power Rule: \(\log_b (M^n) = n \cdot \log_b M\).

Each rule helps us manipulate logarithmic expressions by transforming products, quotients, or powers into simpler additive or subtractive terms. These rules are especially helpful in the process of condensing logarithmic expressions, allowing for more manageable mathematical manipulation.

Logarithmic Expression Condensation

Condensing logarithmic expressions means transforming multiple logarithmic terms into a single, simplified logarithm. This technique leverages the properties of logarithms for efficiency and clarity.

To condense, identify opportunities to combine terms using rules like the product rule or quotient rule.

• Look for repetitions of addition, which can be handled by the product rule.
• Identify subtractions for the application of the quotient rule.

Moreover, don’t forget the power rule, which helps move coefficients as the exponents inside the logarithms. Condensation is useful in equations where it can allow further algebraic steps to solve for variables or equate expressions.

Power Rule for Logarithms

The power rule is a fundamental property used when dealing with logarithmic expressions. It simplifies expressions where the logarithm has a coefficient, by transforming this coefficient into an exponent.

The rule states: \(\log_b (M^n) = n \cdot \log_b M\). This means you can move the coefficient in front of the logarithm as an exponent on its argument.
For example, consider the expression \(3 \ln(x+1)\). Using the power rule, it converts to \(\ln((x+1)^3)\), substantially simplifying the expression.

• This process reduces the complexity of logarithmic expressions.
• It's an efficient way to prepare expressions for further simplification like condensing or expanding.

This rule is crucial whenever you wish to condense or expand logarithmic statements effectively.

Quotient Rule for Logarithms

The quotient rule for logarithms is used when handling two logarithmic terms that involve subtraction. It allows you to condense these terms into a single logarithm showing a division.

The rule formula is: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\).By understanding this rule, you can rewrite expressions with subtracted logarithms as a quotient.
In the exercise, we used this rule to transform \(\ln(x) - \ln((x+1)^3)\) into \(\ln\left(\frac{x}{(x+1)^3}\right)\).

• This transformation aids in simplifying and consolidating the expression.
• It's especially helpful in breaking down algebraic solutions or streamlining computational steps.

By mastering the quotient rule, one gains a significant edge in managing and solving logarithmic-based mathematics problems.