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Chapter 13: Problem 33

Solve each logarithmic equation. $$\log _{11} x=2$$

Short Answer

Expert verified
To solve the logarithmic equation \(\log_{11}x = 2\), we convert it into an exponential equation using the formula \(\log_b a = k\Longleftrightarrow b^k = a\). In this case, the base is 11 and the exponent is 2, so we get \(11^2 = x\). Calculating the value of x, we find that \(x = 121\).

Step by step solution

01

Identifying the base, exponent, and value

In the given equation \(\log_{11} x = 2\), we can easily identify the base as 11, the exponent as 2, and the value of the logarithm as x. Now we need to convert this equation into an exponential equation.
02

Converting logarithmic equation to exponential equation

We'll use the formula \(\log_b a = k\Longleftrightarrow b^k = a\). For our given equation, we have the base as 11, the exponent as 2, and the value of the logarithm as x. So, using the formula, we can rewrite the equation as: \(11^2 = x\)
03

Computing the value of x

Now, we need to calculate the value of x: \(11^2\) equals \(11 \times 11 = 121\). So, we get: \(x = 121\)
04

Conclusion

We've successfully converted the logarithmic equation to an exponential equation and computed the value of x. The solution to the equation \(\log_{11}x = 2\) is \(x = 121\).

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Key Concepts

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

Exponential Equations
An exponential equation is one in which a variable appears as an exponent. These types of equations are crucial in various fields, such as finance, physics, and population studies. Understanding how to manipulate these equations allows us to solve a wide array of real-world problems.

In the context of our log problem, the exponential equation we derived is \(11^2 = x\). This is straightforward. Here, 11 is the base, 2 is the exponent, and \(x\) is the result of raising the base to the exponent. By solving exponential equations, we gain power in calculating growth, decay, and many scientific phenomena.
Base and Exponent Identification
Identifying the base and the exponent is foundational in solving both logarithmic and exponential equations. In any logarithmic form \( \log_b a = k\), the base \(b\) is the number we repeatedly multiply, and \(k\) is the power to which the base is raised to get \(a\).

In the equation \(\log_{11}x = 2\), the base is 11, and the exponent is 2. These components are essential for the next step in solving the equation. Knowing precisely what your base and exponent are simplifies the conversion process to exponential form.
Solving Equations
The process of solving equations involves finding the unknown value that makes the equation true. For our equation \(\log_{11}x = 2\), solving it requires converting it into a more manageable form.
  • First, use the base and exponent identified to set up the equation, as in \(b^k = a\).
  • Then, calculate \(b^k\), which, in our example, means finding \(11^2\).
  • This computation results in \(121\), thus telling us \(x\) equals that value.
The last step confirms that our initial understanding of the logarithmic setup was correct.
Logarithmic to Exponential Conversion
One of the most important techniques in solving logarithmic equations is converting them into exponential forms. This conversion is based on the principle that a logarithm answers the question: "To what power must the base be raised to produce a certain number?"
  • The key formula for conversion is \(\log_b a = k \Leftrightarrow b^k = a\).
  • Applying this to \(\log_{11}x = 2\), convert it to \(11^2 = x\).
This switch not only simplifies the equation but turns it into a directly solvable form, where arithmetic can determine the value easily. Mastering this transformation is elemental in working with logarithmic equations efficiently.

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Most popular questions from this chapter

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