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Chapter 10: Problem 7

Give the value of each expression. $$ 10^{\log \sqrt{3}} $$

Short Answer

Expert verified
The value of the expression \( 10^{\text{log} \sqrt{3}} \) is \sqrt{3} \.

Step by step solution

01

Understand the Logarithmic Property

Recall the logarithmic property that states for any positive number a, and any real number x, the expression \( a^{\text{log}_a x} = x \). This property helps simplify expressions involving logarithms with the same base.
02

Identify the Base

In the given problem, we have \( 10^{\text{log} \sqrt{3}} \). Here, the base is 10, and the logarithm is also in base 10 (implied by the notation \( \text{log} \)).
03

Simplify the Expression

Using the identified logarithmic property, \( 10^{\text{log} \sqrt{3}} = \sqrt{3} \). Thus, the value of the expression simplifies directly.

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Key Concepts

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

headline of the respective core concept
Logarithms are a crucial part of algebra that help transform complex multiplication and division into simpler addition and subtraction problems. A logarithm asks the question, 'To what exponent must we raise a base number to obtain another number?' For example, in the expression \( \log_{a} b \), we're asking for what power we must raise \(a\) to get \(b\). If \(2^{3} = 8\), then \( \log_{2} 8 = 3 \). Using this concept can simplify solving various algebraic equations.
Let's go through this step by step so you can grasp its mechanics smoothly:
  • First, recognize the base in the logarithmic expression, which is often implied to be 10 if not mentioned.
  • Secondly, understand that logarithmic and exponential functions are inverses of each other.
  • Lastly, leverage different logarithmic properties to simplify otherwise complex calculations.
These properties make logarithms very powerful in both theoretical and applied mathematics.
headline of the respective core concept
When dealing with logarithmic expressions, we often see terms like \( \log 100 \) or \( \log x \). Here are some key points to remember:
A logarithmic expression consists of three main components: the base, the argument, and the result. For instance, in \( 10^{\log 100} \), the base is 10, the argument is 100, and the result is 2.
The relationship between exponential and logarithmic forms is given by the equation: \( a^{\log_{a} x} = x \).
  • The base of the logarithm and the base of the exponential term must be the same for this property to hold.
  • If they are the same, logarithmic expressions can easily be simplified.
Practice recognizing these patterns in different problems to become comfortable with switching between exponential and logarithmic forms.
headline of the respective core concept
Simplifying logarithms is a common task in algebra and calculus that makes calculations much easier. The main goal is to reduce complex logarithmic expressions to simpler forms by applying logarithmic properties. In the given exercise, the simplification is straightforward because the base of the logarithm and the exponential are the same.
Let's apply this to the given expression \( 10^{\log \sqrt{3}} \):
  • We identify that 10 is the base of both the logarithm and the exponential term.
  • Using the property \( a^{\log_{a} x} = x \), we can directly simplify it to \( \sqrt{3} \).
This property is particularly useful because it allows us to bypass many intermediate steps. Understanding and applying such properties will make handling logarithmic expressions more efficient and much less intimidating.

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Most popular questions from this chapter

Use the special properties of logarithms to evaluate each expression. $$\log _{6} \frac{1}{6}$$

How long, to the nearest hundredth of a year, would it take \(\$ 4000\) to double at \(3.25 \%\) compounded continuously?

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{21} 0.7496 $$

Solve each equation. Give exact solutions. $$ \log (2 x-1)+\log 10 x=\log 10 $$

Solve each equation. Give exact solutions. $$ \log _{4}(2 x+8)=2 $$
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