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Chapter 1: Problem 12

Simplify. \(10^{\log \sqrt{2}}\)

Short Answer

Expert verified
The simplified form is \(\sqrt{2}\).

Step by step solution

01

Understand the Expression

The expression given is \(10^{\log \sqrt{2}}\). This is an exponentiation where the base is 10, and the exponent is the logarithm of \(\sqrt{2}\).
02

Apply Logarithmic Identity

Recall that an important logarithmic identity is \(a^{\log_b c} = c^{\log_b a}\). In this problem, \(a = 10\), \(b = 10\), and \(c = \sqrt{2}\). Thus, the expression can be rewritten as \(\sqrt{2}^{\log_{10} 10}\).
03

Simplify Using Base Identity

Since \(\log_{10} 10 = 1\), simplify the expression \(\sqrt{2}^{1}\) to simply \(\sqrt{2}\). Therefore, \(10^{\log \sqrt{2}} = \sqrt{2}\).

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Key Concepts

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

Understanding Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number being multiplied, while the exponent specifies how many times the base is used as a factor. Consider the example of exponentiation:
  • For instance, in the expression \(10^3\), 10 is the base, and 3 is the exponent. This means that 10 is multiplied by itself three times, resulting in 1000.
  • Exponentiation is a powerful concept and is fundamental in many areas of mathematics and science. It represents repeated multiplication, offering a concise way to express large numbers.
  • In the context of our exercise, we encounter the base as 10 and the exponent as \(\log \sqrt{2}\). Here, the exponent is a logarithmic expression which we will decode next.
By understanding both the base and the exponent, we can tackle complex expressions like \(10^{\log \sqrt{2}}\).
Logarithmic Identity Simplified
A logarithm is the inverse operation of exponentiation. It answers the question: 'To what exponent must the base be raised to produce a given number?'. Logarithmic identities help in manipulating and simplifying expressions involving logarithms. Let's go through the key identity used here:
  • The crucial identity applied in the given problem is \(a^{\log_b c} = c^{\log_b a}\). This means that an exponent raised to a logarithm can be creatively rearranged where the base and argument of the logarithm are swapped, provided the base remains consistent.
  • In simpler terms, using this identity means we take \(10^{\log_{10} \sqrt{2}}\) and realize it can be rewritten as \(\sqrt{2}^{\log_{10} 10}\).
  • Such rewriting is often advantageous, especially when simplifying based on known logarithmic values, like \(\log_{10} 10\).
Understanding and applying this identity is vital in simplifying expressions involving logarithms, turning complex problems into more manageable forms.
Expression Simplification
Simplifying expressions is about reducing them to their simplest form without changing their value. In our exercise, we initially had the expression \(10^{\log \sqrt{2}}\).
  • After applying the logarithmic identity, we transformed it to \(\sqrt{2}^{\log_{10} 10}\).
  • Recognizing that \(\log_{10} 10\) equals 1 makes this step straightforward; raising a number to power 1 yields the number itself.
  • Thus, \(\sqrt{2}^{1}\) simplifies directly to \(\sqrt{2}\).
  • The original complex-looking expression \(10^{\log \sqrt{2}}\) is now simplified to just \(\sqrt{2}\).
Through understanding exponentiation, applying logarithmic identities, and recognizing simple logarithmic values, we achieve expression simplification. This holistic approach makes tackling seemingly complicated mathematical expressions much easier.

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