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Chapter 11: Problem 6

Reading and Writing. After reading this section, write out the answers to these questions. Use complete sentences. Why is it true that \(\log _{a}(1)=0\) for \(a>0\) and \(a \neq 1 ?\)

Short Answer

Expert verified
Since \(a^0 = 1\) for any \(a>0\) and \(a eq 1\), \(\log _{a}(1)=0\).

Step by step solution

01

- Understand the Logarithmic Property

Recall that the logarithmic function \(\text{log}_a(x)\) is the inverse of the exponential function \(a^y=x\). This means that \(y=\text{log}_a(x)\) is equivalent to \(a^y=x\).
02

- Substitute the Given Values

Substitute \(1\) for \(x\) and \(0\) for \(y\) in the inverse equation. This results in \(a^0=1\).
03

- Apply the Exponential Identity

Use the identity that any non-zero number raised to the power of \(0\) is \(1\): \(a^0=1\). This confirms that the equation holds true when \(y=0\).
04

- State the Conclusion

Since \(a^0=1\) is true for any \(a>0 \) and \(a eq 1\), we confirm that \(\text{log}_a(1)=0\) is always true under these conditions.

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Key Concepts

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

Logarithmic Functions
Logarithmic functions are an important concept in mathematics that help us solve for unknown exponents. The logarithmic function \(\text{log}_a(x)\) is the inverse of an exponential function \(a^y = x\). This means when we see \(\text{log}_a(x)\), we are asking the question: 'To what power must \(a\) be raised, to get \(x\)?' For instance, if we ask what is \(\text{log}_2(8)\), we are asking, 'What power must 2 be raised to, to get 8?' The answer is 3 because \(2^3 = 8\).
Logarithmic functions convert multiplicative relationships into additive ones, which helps in simplifying and solving complex exponential equations.
Inverse Functions
Inverse functions reverse the effect of the original function. If you have a function \(f(x)\), its inverse \(f^{-1}(x)\) undoes what \(f(x)\) does. For exponential and logarithmic functions, this relationship is particularly important.
The exponential function \(a^y = x\) and the logarithmic function \(\text{log}_a(x) = y\) are inverses of each other. This means they undo each other鈥檚 operations, and understanding one helps understand the other. For example, if you know \(\text{log}_a(x) = y\), then \(a^y = x\).
This inverse relationship helps in solving logarithmic equations using exponential techniques and vice versa.
Exponential Identities
Exponential identities are foundational rules in mathematics that describe how exponentiation behaves. One key identity relevant here is that any non-zero number raised to the power of zero is 1. Mathematically, this is stated as \(a^0 = 1\) for any \(a eq 1\).
This identity is crucial in understanding logarithms. For instance, showing \(\text{log}_a(1) = 0\), uses the identity \(a^0 = 1\). Recalling that logarithms are the inverse functions of exponentiation explains why \(\text{log}_a(1) = 0\)
Practicing these identities helps deepen understanding of both exponential and logarithmic properties, making equations easier to handle.
Mathematical Proofs
Mathematical proofs are logical arguments that verify the truth of a mathematical statement. Proofs often use definitions, properties, and logical reasoning. In proving \(\text{log}_a(1) = 0\), we employ the definition of logarithms and the exponential identity \(a^0 = 1\).
Let's break down the proof:
  • Recall the inverse relationship: \(y = \text{log}_a(x)\) implies \a^y = x\
  • Substitute 1 for \(x\) and 0 for \(y\) to get: \(a^0 = 1\)
  • By the exponential identity, \(a^0 = 1\)
Since this holds for any \(a > 0\) and \(a eq 1\), the proof concludes that when \(x = 1\), \(y\) must be 0, hence \(\text{log}_a(1) = 0\).
This step-by-step logical proof illustrates the use of properties and identities in validating mathematical statements.

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