/*! 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 40 Solve the equations. $$ \log... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 40

Solve the equations. $$ \log x=\frac{1}{2} $$

Short Answer

Expert verified
Answer: x = $$\sqrt{10}$$.

Step by step solution

01

Understand the logarithm

The equation we need to solve is: $$\log x = \frac{1}{2}$$. The base of the logarithm is not given, which means it is the common logarithm or base 10. This is represented by: $$\log_{10} x = \frac{1}{2}$$.
02

Rewrite the equation as an exponential equation

To rewrite the equation, recall that $$\log_{a} b = c$$ can be written as $$a^c = b$$. So, using this, rewrite the equation: $$10^{\frac{1}{2}} = x$$.
03

Solve for x

Observe that $$10^{\frac{1}{2}}$$ is the same as $$\sqrt{10}$$. So, the solution for x is: $$x = \sqrt{10}$$.

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Key Concepts

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

Common Logarithm
When we deal with logarithms, if the base number isn't specified, it usually means we are working with a common logarithm. This is a logarithm with base 10, commonly written as \( \log_{10} \) or simply \( \log \).
Common logarithms are used because the base 10 system is familiar to us and aligns with our decimal number system. They allow us to express large and small numbers in a convenient way, especially in scientific contexts.
In the exercise, the use of the common logarithm helped us translate \( \log x = \frac{1}{2} \) into an exponential form, which simplified solving the equation.
Exponential Equations
Exponential equations are equations where variables appear as exponents. These types of equations often look like \( a^x = b \), where \( a \) and \( b \) are constants whereas \( x \) is the variable.
Such equations are used to model a wide range of real-world phenomena because they describe growth and decay processes like population growth or radioactive decay.
In our specific exercise, the logarithmic equation was transformed into terms of an exponential equation, \( 10^{\frac{1}{2}} = x \), allowing us to solve for \( x \). This is a typical method for solving logarithmic equations by converting them into exponential equations.
Solving Exponential Equations
To solve exponential equations, a common approach is to isolate the exponential expression and then apply the inverse operation. Exponential equations involve growth or decay patterns, and understanding this can simplify finding solutions.
  • First, convert the logarithmic form, if present, to an exponential form.
  • Ensure that you simplify the equation for clarity.
  • Solve for the variable directly.
The equation \( 10^{1/2} = x \) can be solved by recognizing that \( 10^{1/2}\) is equivalent to \( \sqrt{10} \).
This direct approach allows us to find \( x \) easily in its simplest form, which is \( \sqrt{10} \). This technique highlights how understanding the reciprocal nature of logarithms and exponentials aids in solving these kinds of problems effectively.

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Most popular questions from this chapter

Rewrite the expression in terms of \(\log A\) and \(\log B\), or state that this is not possible. $$ \log (A \sqrt{B})+\log \left(A^{2}\right) $$

Assume \(a\) and \(b\) are positive constants. Imagine solving for \(x\) (but do not actually do so). Will your answer involve logarithms? Explain how you can tell. $$ 3(\log x)+a=a^{2}+\log x $$

(a) Calculate \(\log 2, \log 20, \log 200\) and \(\log 2000\) and describe the pattern. (b) Using the pattern in part (a) make a guess about the values of \(\log 20,000\) and \(\log 0.2\). (c) Justify the guess you made in part (b) using the properties of logarithms.

Languages diverge over time, and as part of this process, old words are replaced with new ones. \({ }^{13}\) Using methods of glottochronology, linguists have estimated that the number of words on a standardized list of 100 words that remain unchanged after \(t\) millennia is given by $$ f(t)=100 e^{-L t}, \quad L=0.14 $$ Refer to this formula to answer What do your answers tell you about word replacement? Solve \(f(t)=10\)

Write the expressions in the form \(\log _{b} x\) for the given value of \(b\). State the value of \(x\), and verify your answer using a calculator. $$ \frac{\log 90}{4 \log 5}, \quad b=5 $$
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