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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 97
Evaluate or simplify each expression without using a calculator.
Guida verificata passo dopo passo1
Recognize that the expression is \(e^{(\ln 5x^2)}\), which involves the exponential function and the natural logarithm function.
Recall the property of logarithms and exponentials: \(e^{\ln a} = a\) for any positive \(a\).
Apply this property to simplify \(e^{(\ln 5x^2)}\) directly to \$5x^2$.
Note that this simplification holds as long as the argument inside the logarithm, \$5x^2$, is positive, which means \(x \neq 0\).
Therefore, the simplified form of the expression is \$5x^2$.
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Properties of Exponents
Understanding how to manipulate expressions with exponents is essential. For example, the rule e^(a + b) = e^a * e^b allows breaking down complex exponentials, and e^(ln x) simplifies directly to x when x is positive.
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04:06
Rational Exponents
Natural Logarithm (ln) and Its Inverse Relationship with Exponentials
The natural logarithm ln(x) is the inverse function of the exponential function e^x. This means e^(ln y) = y for y > 0, which helps simplify expressions like e^(ln 5x^2) directly to 5x^2.
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5:02Solving Logarithmic Equations
Simplifying Algebraic Expressions
After applying logarithmic and exponential properties, simplifying the resulting algebraic expression is necessary. This includes understanding how to handle powers, coefficients, and variables, such as recognizing that (x^2) remains as is unless further simplification is possible.
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05:07Simplifying Algebraic Expressions
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