/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Find the common logarithm of eac... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 11

Find the common logarithm of each of the given numbers by using a calculator. $$\sqrt{274}$$

Short Answer

Expert verified
The common logarithm of \( \sqrt{274} \) is approximately \( 1.2189 \).

Step by step solution

01

Express the number under the radical

First, express the given number, \( \sqrt{274} \), as a power. Since taking the square root is equivalent to raising a number to the power of \( \frac{1}{2} \), we can rewrite \( \sqrt{274} \) as \( 274^{\frac{1}{2}} \).
02

Apply logarithm to the expression

Apply the common logarithm (base 10) to the number. Using the property of logarithms, \( \log(a^b) = b \cdot \log(a) \), we can express \( \log(\sqrt{274}) \) as \( \frac{1}{2} \cdot \log(274) \).
03

Calculate \( \log(274) \) via calculator

Using a calculator, compute \( \log(274) \). This gives approximately \( \log(274) \approx 2.4378 \).
04

Calculate the common logarithm

Substitute \( \log(274) \) into the expression from Step 2 to find the common logarithm of \( \sqrt{274} \): \[ \log(\sqrt{274}) = \frac{1}{2} \cdot \log(274) \approx \frac{1}{2} \cdot 2.4378 \approx 1.2189 \]

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Key Concepts

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

Logarithm Properties
Logarithms are the inverse operations of exponentiation and hold unique properties that simplify solving many mathematical expressions. A common logarithm is a logarithm with a base of 10. One important property used here is the power rule of logarithms:
  • Power Rule: \( \log(a^b) = b \cdot \log(a) \)
This property was utilized in the exercise to convert the logarithm of a square root into a simpler form. This transformation allows us to multiply the logarithm of the base by the power, effectively simplifying the calculations.
Square Root and Exponents
The concept of square roots and exponents is key to expressing numbers in different forms, thereby facilitating easier computation of their logarithms.
  • A square root, \( \sqrt{x} \), can be expressed as \( x^{\frac{1}{2}} \).
  • Exponentiation indicates repeated multiplication, but when dealing with roots, you translate it into fractional exponents.
Converting \( \sqrt{274} \) to \( 274^{\frac{1}{2}} \) allows further operations with logarithms by applying exponent rules which makes complex problems simpler to handle.
Calculator Usage
To find the common logarithm of numbers swiftly and accurately, calculators become indispensable tools. Here鈥檚 how you can use a calculator effectively:
  • First, enter the number for which you need the logarithm, for example, 274.
  • Then, use the "log" button to calculate \( \log(274) \).
For this problem, a calculator was used to compute \( \log(274) \), which approximately equals 2.4378. Most scientific calculators include this function and provide quick access to these calculations.
Mathematical Expressions
Understanding and manipulating mathematical expressions is essential to solve problems involving logarithms.
  • Start by rewriting the expression in an easier-to-manage form. In this problem, converting the square root into an expression raised to a power is crucial.
  • Apply mathematical operations such as the rules of logarithms to break down complex expressions into manageable calculations.
These steps guide you in transforming and calculating expressions to unravel solutions in complex mathematical problems with ease.

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Most popular questions from this chapter

Find the natural antilogarithms of the given logarithms. $$5.420$$

Use a calculator to solve the given equations. According to one model, the number \(N\) of Americans (in millions) age 65 and older that will have Alzheimer's disease \(t\) years after 2015 is given by \(N=5.1(1.03)^{t} .\) In what year will this number reach 8.0 million?

Use a calculator to solve the given equations. $$9^{x}-3^{x}-12=0$$

Use a calculator to solve the given equations. When a camera flash goes off, the batteries recharge the flash's capacitor to a charge \(Q\) according to \(Q=Q_{0}\left(1-e^{-k t}\right),\) where \(Q_{0}\) is the maximum charge. How long does it take to recharge the capacitor to \(90 \%\) of capacity if \(k=0.5 ?\)

Plot the indicated semilogarithmic graphs for the following application. In a particular electric circuit, called a low-pass filter, the input voltage \(V_{i}\) is across a resistor and a capacitor, and the output voltage \(V_{0}\) is across the capacitor (see Fig. 13.28 ). The voltage gain \(G\) (in \(d B\) ) is given by $$G=20 \log \frac{1}{\sqrt{1+(\omega T)^{2}}}$$ where \(\tan \phi=-\omega T\) Here, \(\phi\) is the phase angle of \(V_{0} / V_{i .}\) For values of \(\omega T\) of 0.01,0.1,0.3 \(1.0,3.0,10.0,30.0,\) and \(100,\) plot the indicated graphs. These graphs are called a Bode diagram for the circuit. Calculate values of \(G\) for the given values of \(\omega T\) and plot a semilogarithmic graph of \(G\) vs. \(\omega T\)
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