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Chapter 10: Problem 55

Solve each equation. $$\log _{x} \frac{1}{25}=-2$$

Short Answer

Expert verified
x = 5

Step by step solution

01

Understand the logarithmic equation

The given equation is \(\text{log}_{x}(\frac{1}{25}) = -2\). This means that x raised to the power of -2 is equal to \(\frac{1}{25}\).
02

Rewrite the equation in exponential form

Rewrite the logarithmic equation \(\text{log}_{x}(\frac{1}{25}) = -2\) as \(x^{-2} = \frac{1}{25}\).
03

Solve for x

Set the equation \(x^{-2} = \frac{1}{25}\) equal to \(x^{-2} = \frac{1}{5^2}\), which implies that \(x^{-2} = 5^{-2} \). Since the exponents and bases are equal, \(x = 5\).

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Key Concepts

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

Understanding Logarithms
Logarithms are the inverse operations of exponentiation. They help us solve equations where the variable is an exponent. For example, the logarithm base 10 of 100 is 2, because 10 raised to the power of 2 equals 100. Mathematically, this is written as \(\text{log}_{10}(100) = 2\). In general, if you have \(\text{log}_{b}(a) = c\), then \(b^c = a\). In our exercise, we need to solve for x in the equation \(\text{log}_{x}(\frac{1}{25}) = -2\). To understand this better, remember that the logarithmic equation \(\text{log}_{x}(y) = z\) translates to \(\text{x^z = y}\) in exponential form.
Exponential Form
Converting a logarithm to its exponential form can make solving the equation easier. In the logarithmic equation \(\text{log}_{x}(\frac{1}{25}) = -2\), we can rewrite this as the exponential form \(\text{x}^{-2} = \frac{1}{25}\). This means that x raised to the power of -2 equals \(\frac{1}{25}\). To simplify further, we need to recognize that \(\frac{1}{25}\) can be written as \((\frac{1}{5})^2\). Therefore, our equation becomes \(\text{x}^{-2} = (\frac{1}{5})^2\).
Solving Logarithmic Equations
Once we have the equation in exponential form, we compare the bases and exponents. From \(\text{x}^{-2} = (\frac{1}{5})^2\), we see that \(\text{x}^{-2} = 5^{-2}\). Since the bases are equal, the exponents must also be equal. Therefore, we conclude that \(\text{x} = 5\). Solving logarithmic equations often involves converting between logarithmic and exponential forms. Remember these steps:
  • Convert the logarithm to its exponential form.
  • Simplify the equation by trying to express both sides with the same base.
  • Compare the exponents and solve for the variable.
Practice will help you get comfortable with these kinds of equations.

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Most popular questions from this chapter

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{1 / 2} 5 $$

Solve each equation. Approximate solutions to three decimal places. $$ 4^{x-2}=5^{3 x+2} $$

Graph each logarithmic function. $$f(x)=\log _{1 / 5} x$$

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

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