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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 97
Chapter 5, Problem 97

Evaluate or simplify each expression without using a calculator. eln5x2e^{\(\ln\)5x^2}

Verified step by step guidance
1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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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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Related Practice
Textbook Question

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