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Chapter 12: Problem 55

Why can't we determine a logarithm of 0? (Hint: Think of the definition of logarithm.)

Short Answer

Expert verified
We can't determine \(\text{log}_b(0)\) because no power of a positive base can give 0.

Step by step solution

01

- Understand the Definition of Logarithm

A logarithm is the power to which a number (the base) must be raised to produce a given number. Mathematically, for a base \(b\) and a number \(x\), the logarithm of \(x\) is defined as \(\text{log}_b(x) = y\) if and only if \(b^y = x\).
02

- Consider a Logarithm with Argument 0

Applying the definition, consider \(\text{log}_b(0)\). According to the definition, this would mean we need to find a \(y\) such that \(b^y = 0\).
03

- Analyze the Exponential Function

For any positive base \(b > 0\), the exponential function \(b^y\) is always positive for all real \(y\). This means \(b^y > 0\) for all real \(y\).
04

- Conclude the Impossibility

Because the exponential function \(b^y\) never reaches 0 for any real number \(y\), there is no real number \(y\) that satisfies \(b^y = 0\). Therefore, \(\text{log}_b(0)\) is undefined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithm definition
A logarithm helps us understand magnitudes by answering the question:

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Most popular questions from this chapter

Find each logarithm. Give approximations to four decimal places. \(\ln 10\)

Use the change-of-base rule (with either common or natural logarithms) to find each logarithm to four decimal places. \(\log _{4} 18\)

Find each logarithm. Give approximations to four decimal places. \(\log 0.1741\)

Find logarithm. Give approximations to four decimal places. \(\log 328.4\)

Find each logarithm. Give approximations to four decimal places. \(\log 0.0326\)
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