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Evaluate or simplify each expression without using a calculator. In e9x
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 95
Solve each equation. 2|ln x|−6=0
Guida verificata passo dopo passo1
Start with the given equation: \(2|\ln x| - 6 = 0\).
Isolate the absolute value expression by adding 6 to both sides: \(2|\ln x| = 6\).
Divide both sides by 2 to solve for the absolute value: \(|\ln x| = 3\).
Recall that \(|A| = B\) means \(A = B\) or \(A = -B\). So, set up two equations: \(\ln x = 3\) and \(\ln x = -3\).
Solve each equation for \(x\) by exponentiating both sides with base \(e\): \(x = e^{3}\) and \(x = e^{-3}\). Remember to check that \(x > 0\) since the natural logarithm is only defined for positive \(x\).
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Properties of Logarithmic Functions
The natural logarithm function, denoted ln(x), is defined only for positive values of x. Understanding its domain and behavior is essential when solving equations involving ln(x), as it restricts the possible solutions to x > 0.
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Graphs of Logarithmic Functions
Absolute Value Equations
An absolute value equation like |A| = B splits into two cases: A = B and A = -B. This concept is crucial for solving equations involving absolute values, as it allows us to consider both positive and negative scenarios of the expression inside the absolute value.
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06:00Categorizing Linear Equations
Solving Exponential and Logarithmic Equations
To solve equations involving logarithms, we often isolate the logarithmic expression and then exponentiate both sides to eliminate the log. This process helps convert the equation into a more manageable algebraic form for finding the variable.
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5:02Solving Logarithmic Equations
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