Find The Value Of { X$}$ In The Equation { X = \log_{25}(5)$}$.A. { \frac{1}{5}$}$ B. 5 C. 2 D. { \frac{1}{2}$}$
Introduction
Logarithmic equations can be challenging to solve, but with a clear understanding of the properties of logarithms, we can break them down into manageable steps. In this article, we will focus on solving the equation {x = \log_{25}(5)$}$, which involves finding the value of {x$}$ in a logarithmic expression.
Understanding Logarithms
Before we dive into solving the equation, let's review the basics of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if {a^b = c$}$, then {\log_a(c) = b$}$. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number.
The Equation {x = \log_{25}(5)$}$
Now that we have a basic understanding of logarithms, let's examine the equation {x = \log_{25}(5)$}$. This equation asks us to find the value of {x$}$ such that ${25^x = 5\$}.
Step 1: Simplify the Equation
To simplify the equation, we can use the property of logarithms that states {\log_a(b) = \frac{1}{\log_b(a)}$}$. Applying this property to the equation, we get:
{x = \log_{25}(5) = \frac{1}{\log_5(25)}$}$
Step 2: Evaluate the Logarithm
Now that we have simplified the equation, let's evaluate the logarithm {\log_5(25)$}$. Since ${5^2 = 25\$}, we can conclude that {\log_5(25) = 2$}$.
Step 3: Substitute the Value
Substituting the value of {\log_5(25)$}$ into the equation, we get:
{x = \frac{1}{2}$}$
Conclusion
In conclusion, the value of {x$}$ in the equation {x = \log_{25}(5)$}$ is {\frac{1}{2}$}$. This solution is based on the properties of logarithms and the simplification of the equation.
Answer
The correct answer is:
- D. {\frac{1}{2}$}$
Why is this the correct answer?
This is the correct answer because it satisfies the equation {x = \log_{25}(5)$}$. By substituting {\frac{1}{2}$}$ into the equation, we get ${25^{\frac{1}{2}} = 5\$}, which is true.
What is the significance of this problem?
This problem is significant because it demonstrates the importance of understanding logarithmic properties and how to apply them to solve equations. By mastering these concepts, we can tackle a wide range of mathematical problems and develop a deeper understanding of mathematical relationships.
Real-World Applications
Logarithmic equations have numerous real-world applications, including:
- Finance: Logarithmic equations are used to calculate interest rates, investment returns, and other financial metrics.
- Science: Logarithmic equations are used to model population growth, chemical reactions, and other scientific phenomena.
- Engineering: Logarithmic equations are used to design and optimize systems, such as electronic circuits and mechanical systems.
Conclusion
Introduction
Logarithmic equations can be challenging to solve, but with a clear understanding of the properties of logarithms, we can break them down into manageable steps. In this article, we will provide a Q&A guide to help you understand and solve logarithmic equations.
Q: What is a logarithmic equation?
A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if {a^b = c$}$, then {\log_a(c) = b$}$.
Q: How do I simplify a logarithmic equation?
A: To simplify a logarithmic equation, you can use the property of logarithms that states {\log_a(b) = \frac{1}{\log_b(a)}$}$. You can also use the property that states {\log_a(b^c) = c \cdot \log_a(b)$}$.
Q: What is the difference between a logarithmic equation and an exponential equation?
A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, {x = \log_{25}(5)$}$ is a logarithmic equation, while ${25^x = 5\$} is an exponential equation.
Q: How do I solve a logarithmic equation?
A: To solve a logarithmic equation, you can follow these steps:
- Simplify the equation using the properties of logarithms.
- Evaluate the logarithm.
- Substitute the value into the equation.
- Solve for the variable.
Q: What is the value of {x$}$ in the equation {x = \log_{25}(5)$}$?
A: The value of {x$}$ in the equation {x = \log_{25}(5)$}$ is {\frac{1}{2}$}$. This is because ${25^{\frac{1}{2}} = 5\$}, which satisfies the equation.
Q: How do I apply logarithmic equations in real-world situations?
A: Logarithmic equations have numerous real-world applications, including:
- Finance: Logarithmic equations are used to calculate interest rates, investment returns, and other financial metrics.
- Science: Logarithmic equations are used to model population growth, chemical reactions, and other scientific phenomena.
- Engineering: Logarithmic equations are used to design and optimize systems, such as electronic circuits and mechanical systems.
Q: What are some common mistakes to avoid when solving logarithmic equations?
A: Some common mistakes to avoid when solving logarithmic equations include:
- Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
- Not evaluating the logarithm: Failing to evaluate the logarithm can lead to incorrect solutions.
- Not substituting the value: Failing to substitute the value into the equation can lead to incorrect solutions.
Conclusion
In conclusion, solving logarithmic equations requires a clear understanding of logarithmic properties and the ability to apply them to simplify and solve equations. By mastering these concepts, we can tackle a wide range of mathematical problems and develop a deeper understanding of mathematical relationships.
Additional Resources
For more information on logarithmic equations, check out the following resources:
- Mathway: A math problem solver that can help you solve logarithmic equations.
- Khan Academy: A free online resource that provides video lessons and practice exercises on logarithmic equations.
- Wolfram Alpha: A computational knowledge engine that can help you solve logarithmic equations and other mathematical problems.