Textbook Question
In Exercises 13–15, write each equation in its equivalent exponential form. 1/2 = log49 7
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 5, Problem 13
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln(e2/5)
Verified step by step guidance1
Recall the logarithmic identity that relates the natural logarithm and exponentials: \(\ln\left(e^x\right) = x\). This means the natural logarithm of \(e\) raised to any power simplifies directly to that power.
Identify the exponent in the given expression \(\ln\left(e^{2/5}\right)\). Here, the exponent is \(\frac{2}{5}\).
Apply the identity by rewriting the expression as \(\ln\left(e^{2/5}\right) = \frac{2}{5}\).
Since the expression simplifies directly to a number, no further expansion is needed.
Thus, the expanded form of \(\ln\left(e^{2/5}\right)\) is simply the exponent \(\frac{2}{5}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Properties of logarithms include rules such as the product, quotient, and power rules that simplify logarithmic expressions. For example, the power rule states that ln(a^b) = b * ln(a), which is essential for expanding and simplifying expressions like ln(e^(2/5)).
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Change of Base Property
Natural Logarithm and the Number e
The natural logarithm, denoted ln, is the logarithm with base e, where e is approximately 2.718. A key property is that ln(e^x) = x, which allows direct simplification of expressions involving e raised to a power inside a logarithm.
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Evaluating Logarithmic Expressions Without a Calculator
Some logarithmic expressions can be simplified exactly using known properties and values, avoiding the need for a calculator. Recognizing forms like ln(e^x) helps evaluate expressions quickly and accurately by reducing them to their exponents.
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Related Practice
Textbook Question
Write each equation in its equivalent exponential form. log3 81 = y
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Textbook Question
Write each equation in its equivalent logarithmic form. ∛8 = 2
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Textbook Question
Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. g(x) = (3/2)x
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Textbook Question
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 31-x=1/27
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Textbook Question
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 6(x−3)/4=√6
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