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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 37
Chapter 5, Problem 37

Evaluate each expression without using a calculator. log4 1

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1
Recall the definition of logarithm: \(\log_b a = c\) means that \(b^c = a\).
In this problem, we have \(\log_4 1\), so we want to find the exponent \(c\) such that \$4^c = 1$.
Remember that any nonzero number raised to the power of 0 equals 1, i.e., \(b^0 = 1\) for \(b \neq 0\).
Since \$4^0 = 1$, it follows that \(\log_4 1 = 0\).
Therefore, the value of \(\log_4 1\) is the exponent that makes the base 4 equal to 1, which is 0.

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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 1

For any positive base b (b ≠ 1), log_b(1) is always 0 because b raised to the power 0 equals 1. This property simplifies evaluating logarithms where the argument is 1.
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Properties of Logarithms Without a Calculator

Evaluating logarithms without a calculator relies on recognizing special values and applying logarithmic properties, such as log_b(b) = 1 and log_b(1) = 0. Familiarity with these properties allows quick simplification.
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Change of Base Property
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. 7(x+2)=410

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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. ln(x3x2+1(x+1)4)\(\ln\) \(\left\)( \(\frac{x^3 \sqrt{x^2 + 1}\)}{(x + 1)^4} \(\right\))

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Textbook Question

Evaluate each expression without using a calculator. log5 5

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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+2

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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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