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Chapter 4: Problem 2

Simplify the following expressions. $$\ln x^{5}-\ln x^{3}$$

Short Answer

Expert verified
x^{5} - x^{3} = x^{2}

Step by step solution

01

- Apply Logarithmic Property

Use the property of logarithms that states \(\ \ln a - \ln b = \ln \left( \frac{a}{b} \right)\ \). In this case, let \ a = x^{5} \ and \ b = x^{3} \: \( x^{5} - x^{3} = \frac{x^{5}}{x^{3}}\)
02

- Simplify the Fraction

Simplify the fraction \ \ \frac{x^{5}}{x^{3}}\, using the rule for exponents that states \ \ x^{a}/x^{b} = x^{a-b}\: \( \frac{x^{5}}{x^{3}} = x^{5-3} = x^{2}\)
03

- Final Simplification

The expression is now simplified. So, \( x^{5} - x^{3} = x^{2}\)

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Key Concepts

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

logarithmic properties
Logarithmic properties are rules that help manipulate and simplify logarithmic expressions. They are essential to master for solving logarithmic equations. One key property used in simplifications is:
  • The Difference Rule: \(\boldsymbol{\text{ln}(a) - \text{ln}(b) = \text{ln}\bigg(\frac{a}{b}\bigg)}}\)
This rule lets you combine two logarithms into one by turning a subtraction into a division inside the logarithm. In the given expression \(\boldsymbol{\text{ln}(x^{5}) - \text{ln}(x^{3})}}\), applying the Difference Rule gives us: \(\text{ln}\bigg(\frac{x^{5}}{x^{3}}\bigg)\).
These properties streamline calculations, letting you simplify complex expressions step by step.
exponent rules
Exponent rules define how to handle powers and simplify expressions involving exponents. Understanding these rules is vital for simplifying logarithmic expressions.
  • Division Rule: \( \boldsymbol{\frac{x^{a}}{x^{b}} = x^{a-b}} \)
In the given problem, after applying the logarithmic property, the expression becomes \( \text{ln}\bigg(\frac{x^{5}}{x^{3}}\bigg) \). We use the Division Rule to simplify \( \frac{x^{5}}{x^{3}} \) as follows: \( x^{5-3} = x^{2} \).
Mastery of these exponent rules allows you to effortlessly handle any powers or roots within expressions, making the process of simplification straightforward.
simplifying expressions
Simplifying expressions involves breaking down complex mathematical statements into their simplest form. This often requires a combination of several mathematical rules.
  • First, apply logarithmic properties to combine or split logarithms.
  • Next, utilize exponent rules to simplify any powers.
In our example:
1. We started with \( \boldsymbol{\text{ln}(x^{5}) - \text{ln}(x^{3})} \).
2. Applied the Difference Rule to get \( \text{ln}\bigg(\frac{x^{5}}{x^{3}}\bigg)\).
3. Simplified \( \frac{x^{5}}{x^{3}} \) using exponent rules, yielding \( x^{2} \).
4. Ending up with the simplified expression \( \boldsymbol{\text{ln}(x^{2})} \).
By following these steps, we ensure every part of the expression is simplified. This process makes complex problems much more manageable.

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