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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 95
Chapter 5, Problem 95

Evaluate or simplify each expression without using a calculator. In e9x

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1
Identify the expression given: \(e^{9x}\). This is an exponential expression where the base is the constant \(e\) (Euler's number) and the exponent is \$9x$.
Recall the properties of exponents, especially for exponential functions: \(e^{a+b} = e^a \cdot e^b\) and \((e^a)^b = e^{ab}\). These can help simplify expressions involving \(e^{9x}\) if combined with other terms.
If the problem involves simplifying or rewriting \(e^{9x}\), consider if it can be expressed as \((e^x)^9\) by using the power of a power rule: \((e^x)^9 = e^{9x}\).
If the expression is part of a larger problem (like addition, subtraction, or multiplication with other exponential terms), apply the appropriate exponent rules to combine or simplify terms.
Since no calculator is allowed, leave the expression in exponential form or simplify using algebraic manipulation without evaluating the numerical value of \(e\).

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Key Concepts

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

Exponential Functions

Exponential functions involve expressions where a constant base is raised to a variable exponent, such as e^(9x). Understanding their properties, like growth behavior and how to manipulate exponents, is essential for simplifying or evaluating these expressions.
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Properties of Exponents

The properties of exponents include rules like a^(m+n) = a^m * a^n and (a^m)^n = a^(mn). These rules allow for the simplification and evaluation of exponential expressions without a calculator by combining or breaking down exponents.
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Natural Exponential Base (e)

The number e (~2.718) is the base of natural logarithms and exponential functions. Recognizing e as a special constant helps in understanding continuous growth models and simplifies working with expressions like e^(9x) in algebra.
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Related Practice
Textbook Question

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln x + ln(2x) = ln(3x)

Textbook Question

Solve each equation. 2|ln x|−6=0

Textbook Question

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(x + 1) = ln x + ln 1

Textbook Question

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(5x) + ln 1 = ln(5x)

Textbook Question

Find all zeros of f(x) = x³ + 5x² – 8x + 2.

Textbook Question

Solve each equation. 3|log x|−6=0

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