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16. Condense the logarithms: a) 4logx - 2logx b) 5logx + 3logy + 2logy + 7logz

16. Condense the logarithms: a) 4logx - 2logx b) 5logx + 3logy + 2logy + 7logz

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Logarithm and Exponential Simplification Exercises** In this section, we will cover problems related to condensing logarithmic expressions as well as simplifying exponential expressions. Understanding these concepts is crucial for a solid foundation in algebra and pre-calculus. ### Problem 16: Condense the logarithms Condensing logarithms involves combining multiple logarithmic terms into a single logarithm. Here are the given expressions: a) \(4 \log x - 2 \log x\) b) \(5 \log x + 3 \log y + 2 \log y + 7 \log z\) ### Problem 17: Simplify the Exponential Expression Simplifying exponential expressions involves using properties of exponents to rewrite the expression in a simpler form. Here is the given expression: \(\ln e^{2x}\) #### Explanation of Concepts: 1. **Logarithm Properties:** - **Product Property**: \(\log_b (mn) = \log_b (m) + \log_b (n)\) - **Quotient Property**: \(\log_b \left(\frac{m}{n}\right) = \log_b (m) - \log_b (n)\) - **Power Property**: \(\log_b (m^n) = n \log_b (m)\) 2. **Natural Logarithm Properties:** - The natural logarithm, denoted \(\ln\), has a base of \(e\) (where \(e \approx 2.718\)). - **Exponential Property**: \(\ln (e^x) = x\) By applying these properties, the logarithmic and exponential expressions can be simplified effectively.
Transcribed Image Text:**Logarithm and Exponential Simplification Exercises** In this section, we will cover problems related to condensing logarithmic expressions as well as simplifying exponential expressions. Understanding these concepts is crucial for a solid foundation in algebra and pre-calculus. ### Problem 16: Condense the logarithms Condensing logarithms involves combining multiple logarithmic terms into a single logarithm. Here are the given expressions: a) \(4 \log x - 2 \log x\) b) \(5 \log x + 3 \log y + 2 \log y + 7 \log z\) ### Problem 17: Simplify the Exponential Expression Simplifying exponential expressions involves using properties of exponents to rewrite the expression in a simpler form. Here is the given expression: \(\ln e^{2x}\) #### Explanation of Concepts: 1. **Logarithm Properties:** - **Product Property**: \(\log_b (mn) = \log_b (m) + \log_b (n)\) - **Quotient Property**: \(\log_b \left(\frac{m}{n}\right) = \log_b (m) - \log_b (n)\) - **Power Property**: \(\log_b (m^n) = n \log_b (m)\) 2. **Natural Logarithm Properties:** - The natural logarithm, denoted \(\ln\), has a base of \(e\) (where \(e \approx 2.718\)). - **Exponential Property**: \(\ln (e^x) = x\) By applying these properties, the logarithmic and exponential expressions can be simplified effectively.
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