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Chapter 4: Problem 77

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

Short Answer

Expert verified
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 by step solution

01

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: \(\ln(x) - \ln{(x+1)^3}\)
02

Applying the quotient rule

Recognize any chances to apply the quotient rule. This rule can be applied when there鈥檚 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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檛 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.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\log _{10} 8 x-\log _{10}(1+\sqrt{x})=2$$

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\log _{10} x=(x-3)^{2}$$

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln 4 x=2.1$$

Use the regression feature of a graphing utility to find a logarithmic model \(y=a+b \ln x\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(1, 8.5), (2, 11.4), (4, 12.8), (6, 13.6), (8, 14.2),(10, 14.6)$$

Factor the polynomial. $$2 x^{3}+x^{2}-45 x$$
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