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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 25
Chapter 5, Problem 25

In Exercises 19–29, evaluate each expression without using a calculator. If evaluation is not possible, state the reason. ln e5

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Recall the property of logarithms that states: \(\ln\left(e^x\right) = x\). This is because the natural logarithm function \(\ln(x)\) is the inverse of the exponential function \(e^x\).
Identify the expression inside the logarithm: \(e^5\). Here, the exponent is 5.
Apply the property directly to simplify the expression: \(\ln\left(e^5\right) = 5\).
Since the logarithm and the exponential functions are inverses, the expression simplifies exactly to the exponent without any further calculation.
Therefore, the value of \(\ln\left(e^5\right)\) is simply 5.

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Key Concepts

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

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the inverse function of the exponential function with base e. It answers the question: to what power must e be raised to get a certain number? For example, ln(e^x) = x.
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The Natural Log

Exponential Function with Base e

The exponential function e^x involves the constant e (approximately 2.718), raised to the power x. It is a fundamental function in algebra and calculus, often used to model growth or decay processes.
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The Number e

Inverse Properties of Logarithms and Exponentials

Logarithms and exponentials are inverse operations, meaning ln(e^x) = x and e^(ln x) = x for x > 0. This property allows simplification of expressions involving ln and e without a calculator.
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Logarithms Introduction
Related Practice
Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 10x=3.91

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Textbook Question

Evaluate each expression without using a calculator. log3 27

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Textbook Question

Evaluate each expression without using a calculator. log5 (1/5)

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Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log6 (36/(√(x+1))

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Textbook Question

Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = 2x+1

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Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ex=5.7

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