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Chapter 4: Problem 122

Explain why the logarithm of 1 with base \(b\) is \(0 .\)

Short Answer

Expert verified
The logarithm of \(1\) to any base \(b\) is \(0\) because any number raised to the power of \(0\) is \(1\), which is a fundamental property of logarithms and exponentiation.

Step by step solution

01

Understanding Logarithms

Logarithm is the inverse operation to exponentiation, just as subtraction is the inverse of addition and division is the inverse of multiplication. It answers the question how many of one number do we multiply to get another number. In other words, in \(log_b (a) = n, b^n = a\). This will be crucial in understanding why the logarithm of 1 for any base \(b\) is \(0\).
02

Applying Logarithm basics

For any base \(b\), when we are calculating \(log_b (1)\), what we are essentially asking is, 'what power should we raise \(b\) to get \(1\)?' The only power that we can raise any real number to, in order to result in \(1\), is \(0\). Therefore, \(b^0 = 1\), and hence, by definition, \(log_b (1) = 0\).
03

Concluding the proof

Using basic logarithm properties and the fact that raising any number to the power of \(0\) yields \(1\), we have determined that the logarithm of \(1\) to any base \(b\) is \(0\). This is fundamental to the properties and rules of logarithm.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations $$ \log (3 x+1)=5 \text { and } \log (3 x+1)=\log 5 $$ are similar, I solved them using the same method.

Find the domain of each logarithmic function. $$ f(x)=\log _{5}(x+6) $$

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{3}(x-1)=2 $$

Evaluate or simplify each expression without using a calculator. $$ \log 1000 $$

Evaluate or simplify each expression without using a calculator. $$ \ln \frac{1}{e^{6}} $$
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