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Chapter 1: Problem 8

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

Short Answer

Expert verified
\( \log \sqrt{10} \) simplifies to \( \frac{1}{2} \log 10 \).

Step by step solution

01

Understand the Problem

We are asked to simplify the expression \( \log \sqrt{10} \). In this case, \( \log \) represents either the logarithm with base 10 or natural logarithm, depending on context, but the simplification process will be the same.
02

Rewrite the Square Root as an Exponent

Recall that the square root of a number can be expressed as an exponent of \( \frac{1}{2} \). Thus, \( \sqrt{10} \) can be rewritten as \( 10^{1/2} \).
03

Use the Logarithmic Power Rule

Apply the power rule for logarithms, which states that \( \log(a^b) = b \cdot \log(a) \). Here, \( a = 10 \) and \( b = \frac{1}{2} \). Therefore, \( \log(10^{1/2}) = \frac{1}{2} \cdot \log(10) \).
04

Simplify the Expression (If Log Base 10)

Assuming the log base is 10, \( \log_{10}(10) = 1 \). Thus, \( \frac{1}{2} \cdot \log_{10}(10) = \frac{1}{2} \times 1 = \frac{1}{2} \).
05

Simplify the Expression (If Natural Log)

If it is a natural log, then the simplified expression is \( \frac{1}{2} \ln(10) \). Without specifying the base, both simplified forms are technically correct.

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Key Concepts

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

Simplifying Expressions
When it comes to simplifying expressions, especially log expressions, the goal is to reduce it to its most basic form. To handle the expression \( \log \sqrt{10} \), we start by understanding the relationship between the square root and exponents.
  • Begin by transforming the square root into an exponent. The square root of any number, \( x \), is equivalent to \( x^{1/2} \).
  • This means, instead of \( \sqrt{10} \), we write \( 10^{1/2} \). This change allows us to use logarithmic rules more effectively.
By rewriting the problem and recognizing its components, you simplify expressing even complex log equations.
Logarithmic Rules
Knowing how logarithms work and the rules that govern them is vital for simplifying log expressions. Let's focus on the logarithmic power rule for this example.
  • The power rule states: \( \log(a^b) = b \cdot \log(a) \). This rule enables you to pull exponents out in front of the log, making calculations easier.
  • Applying it to \( \log(10^{1/2}) \), we derive \( \frac{1}{2} \cdot \log(10) \).
It's crucial to remember that using logarithmic rules accurately transforms complex expressions into simpler, solvable forms.
Exponents
Exponents play a key role in both logarithms and simplifying expressions. They tell you how many times to multiply a number by itself.
  • The expression \( \sqrt{10} \) becomes \( 10^{1/2} \) using the general rule that \( x^{1/n} \) represents the nth root of \( x \). This applies to simplify roots into exponents.
  • Recognizing and transforming roots into exponential form is an essential step when working with logarithms, as it directly impacts how log rules apply.
With a clear understanding of exponents, you can dissect and reconstruct expressions, making the entire logarithmic simplification process a breeze.

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Most popular questions from this chapter

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