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Ch. 4 - Exponential and Logarithmic Functions

Blitzer - College Algebra 8th Edition

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

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Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

문제 37

5장, 문제 37

Evaluate each expression without using a calculator. log₄ 1

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

49s

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

추천 영상:

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.

추천 영상:

Logarithms Introduction

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.

추천 영상:

Change of Base Property

관련 실천

교과서 질문

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

교과서 질문

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\(\ln\) \(\left\)( \(\frac{x^3 \sqrt{x^2 + 1}\)}{(x + 1)^4} \(\right\))$

교과서 질문

The figure shows the graph of f(x) = e^x. 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 h(x) = e^x-1+2

교과서 질문

The figure shows the graph of f(x) = e^x. 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) = e^x+2

교과서 질문

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)

교과서 질문

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. 7^0.3x=813