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Chapter 12: Problem 126

Solve. $$ \left|\log _{3} x\right|=2 $$

Short Answer

Expert verified
The solutions are \(x = 9\) and \(x = \frac{1}{9}\).

Step by step solution

01

Understand the absolute value equation

The problem \(\textstyle\left|\log _{3} x\right|=2\) involves an absolute value, which means \(\log_{3} x\) can either be 2 or -2.
02

Set up two equations

This absolute value equation results in two separate equations: \(\log_{3} x = 2\) and \(\log_{3} x = -2\).
03

Solve the first equation

To solve \(\log_{3} x = 2\), convert it to exponential form: \(3^{2} = x\). Hence, \(x = 9\).
04

Solve the second equation

To solve \(\log_{3} x = -2\), convert it to exponential form: \(3^{-2} = x\). Hence, \(x = \frac{1}{9}\).
05

Combine the solutions

The solutions to the equation \(\textstyle\left|\log _{3} x\right|=2\) are \(x = 9\) and \(x = \frac{1}{9}\).

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Key Concepts

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

absolute value
In mathematics, the absolute value of a real number is its distance from zero on the number line, regardless of direction. This means that the absolute value of both 3 and -3 is 3. When solving equations that involve absolute values, you must consider both the positive and negative scenarios. In the context of the original exercise, \(\textstyle\left|\log _{3} x\right|=2\) translates to two possible equations: \(\log_{3} x = 2\) and \(\log_{3} x = -2\). Each scenario is important to get all possible solutions.
exponential form
Exponential form is a way of expressing mathematical expressions involving log functions as exponents. It is useful for solving logarithmic equations. The general relationship between logarithms and exponents is given by \(a^{b} = c\) if and only if \(\log_{a} c = b\).
  • For \(\log_{3} x = 2\), you convert it to exponential form to get \(3^{2} = x\), which simplifies to \(x = 9\).
  • Similarly, for \(\log_{3} x = -2\), you convert it to exponential form \(3^{-2} = x\), which simplifies to \(x = \frac{1}{9}\).

By converting to exponential form, we can straightforwardly solve for \(x\).
logarithmic functions
Logarithmic functions are the inverses of exponential functions, providing a powerful tool in mathematics to solve for unknown exponents. The function \(\log_{a} (x)\) represents the exponent to which a base \(a\) must be raised to produce the number \(x\).
For instance, \(\log_{3} (9) = 2\) because \(3^{2} = 9\). Understanding this relationship allows for converting between logarithmic and exponential forms efficiently.
Additionally, logarithmic properties facilitate equation solving:
  • \textbf{Product Property:} \(\log_{a} (xy) = \log_{a} (x) + \log_{a} (y)\)
  • \textbf{Quotient Property:} \(\log_{a} \left(\frac{x}{y}\right) = \log_{a} (x) - \log_{a} (y)\)
  • \textbf{Power Property:} \(\log_{a} (x^{n}) = n\cdot\log_{a} (x)\)
Using these properties, you can simplify and solve a wide range of logarithmic equations.

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Most popular questions from this chapter

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b} 45$$

How would you explain to a classmate why \(\log _{2} 5=\log 5 / \log 2\) and \(\log _{2} 5=\ln 5 / \ln 2 ?\)

Given \(\log _{b} 3=0.792 \text { and } \log _{b} 5=1.161\). If possible, use the properties of logarithms to calculate numerical values for each of the following. $$\log _{b} \frac{3}{5}$$

Use the properties of logarithms to find each of the following. $$\log _{2} 16^{5}$$

Solve. $$ \log _{32} x=\frac{2}{3} $$
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