Textbook Question
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.
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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 35
Evaluate each expression without using a calculator. log5 5
Verified step by step guidance1
Recall the definition of a logarithm: \(\log_b a = c\) means that \(b^c = a\).
In this problem, we have \(\log_5 5\), which asks: "To what power must 5 be raised to get 5?"
Since \$5^1 = 5$, the exponent that satisfies this equation is 1.
Therefore, \(\log_5 5 = 1\) because the base and the argument are the same.
This is a general property of logarithms: \(\log_b b = 1\) for any positive base \(b \neq 1\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Logarithms
A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition is essential to evaluate logarithmic expressions.
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Logarithms Introduction
Logarithm of a Base to Itself
The logarithm of a base raised to itself, such as log_b(b), always equals 1 because the base raised to the power 1 equals itself. This property simplifies expressions like log5 5 directly to 1.
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7:30Logarithms Introduction
Properties of Logarithms
Logarithms follow specific properties, such as log_b(b^x) = x and log_b(1) = 0. Recognizing these properties helps in simplifying and evaluating logarithmic expressions without a calculator.
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5:36Change of Base Property
Related Practice
Textbook Question
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.
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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. e(5x−3) - 2 =10,476
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Textbook Question
Evaluate each expression without using a calculator. log4 1
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Textbook Question
The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = ex-1
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Textbook Question
In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log2 (x-2)
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