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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 1.3.19
Chapter 1, Problem 1.3.19

Evaluate each expression without a calculator.


a. log₁₀ 1000

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1
Identify the base of the logarithm. In this case, the base is 10, as indicated by the subscript in log₁₀.
Recall the definition of a logarithm: \( \log_b(a) = c \) means \( b^c = a \).
Apply the definition to the given expression: \( \log_{10}(1000) = c \) means \( 10^c = 1000 \).
Recognize that 1000 can be expressed as a power of 10: \( 1000 = 10^3 \).
Conclude that since \( 10^c = 10^3 \), it follows that \( c = 3 \).

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Key Concepts

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

Logarithms

Logarithms are the inverse operations of exponentiation. The logarithm of a number is the exponent to which a base must be raised to produce that number. For example, in the expression log₁₀ 1000, we are looking for the power to which 10 must be raised to equal 1000.
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Logarithms Introduction

Base of a Logarithm

The base of a logarithm indicates the number that is raised to a power. In log₁₀ 1000, the base is 10. Understanding the base is crucial because it determines the scale of the logarithmic function and how the values relate to one another.
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Properties of Logarithms

Logarithms have several key properties that simplify calculations. One important property is that logₐ (b * c) = logₐ b + logₐ c, which allows for the breaking down of complex logarithmic expressions. This property can be useful in evaluating logarithmic expressions without a calculator.
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Related Practice
Textbook Question

Solving equations Solve each equation.

√2 sin 3Θ + 1 = 2, 0 ≤ Θ ≤ π

Textbook Question

Find the inverse f−1(x)f^{-1}\(\left\)(x\(\right\)) of each function (on the given interval, if specified).

f(x)=10−2xf\(\left\)(x\(\right\))=10^{-2x}

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Textbook Question

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places.


log660\(\log\)_660

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Textbook Question

State whether the functions represented by graphs A , B , C and in the figure are even, odd, or neither. <IMAGE>

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Textbook Question

Finding inverses Find the inverse function.


ƒ(x) = 3x - 4

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Textbook Question

Properties of logarithms Assume logbx = 0.36, logby= 0.56 and logbz = 0.83 . Evaluate the following expressions.


logbx²

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