Select The Correct Answer.What Is The Value Of $x$ In $2\left(5^x\right)=14$?A. $ Log 2 \log 2 Lo G 2 [/tex] B. $\frac{\log 7}{\log 5}$ C. $\frac{\log 5}{\log 7}$ D. $ Log 7 − Log 5 \log 7-\log 5 Lo G 7 − Lo G 5 [/tex]
Introduction
Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithmic functions. In this article, we will explore how to solve exponential equations, focusing on the specific problem of finding the value of $x$ in the equation $2\left(5^x\right)=14$. We will break down the solution into manageable steps, using logarithmic properties to isolate the variable $x$.
Understanding Exponential Equations
An exponential equation is an equation in which the variable is raised to a power. In the given equation, $2\left(5^x\right)=14$, the variable $x$ is raised to the power of 5. To solve for $x$, we need to isolate the variable on one side of the equation.
Step 1: Divide Both Sides by 2
The first step in solving the equation is to divide both sides by 2, which will eliminate the coefficient 2 on the left-hand side of the equation.
Simplifying the equation, we get:
Step 2: Take the Logarithm of Both Sides
To isolate the variable $x$, we need to take the logarithm of both sides of the equation. We will use the logarithmic property that states $\log_a(b^c)=c\log_a(b)$.
Taking the logarithm of both sides, we get:
Using the logarithmic property, we can rewrite the equation as:
Step 3: Divide Both Sides by log(5)
To isolate the variable $x$, we need to divide both sides of the equation by $\log(5)$.
Simplifying the equation, we get:
Conclusion
In this article, we have solved the exponential equation $2\left(5^x\right)=14$ using logarithmic properties. We have broken down the solution into manageable steps, isolating the variable $x$ on one side of the equation. The final answer is:
This solution demonstrates the importance of logarithmic functions in solving exponential equations. By using logarithmic properties, we can isolate the variable and find the value of $x$.
Answer Options
Based on the solution, we can evaluate the answer options:
A. $\log 2$: This is not the correct answer, as the solution involves the logarithm of 7 and 5, not 2.
B. $\frac{\log 7}{\log 5}$: This is the correct answer, as it matches the solution we obtained.
C. $\frac{\log 5}{\log 7}$: This is not the correct answer, as the solution involves the logarithm of 7 in the numerator and 5 in the denominator.
D. $\log 7-\log 5$: This is not the correct answer, as the solution involves the logarithm of 7 and 5 in the numerator and denominator, not a subtraction.
Final Answer
The final answer is:
Introduction
In our previous article, we explored how to solve exponential equations using logarithmic properties. We focused on the specific problem of finding the value of $x$ in the equation $2\left(5^x\right)=14$. In this article, we will continue to provide a Q&A guide on solving exponential equations.
Q: What is an exponential equation?
A: An exponential equation is an equation in which the variable is raised to a power. For example, $2\left(5^x\right)=14$ is an exponential equation, where the variable $x$ is raised to the power of 5.
Q: How do I solve an exponential equation?
A: To solve an exponential equation, you need to isolate the variable on one side of the equation. You can do this by using logarithmic properties, such as the property that states $\log_a(b^c)=c\log_a(b)$.
Q: What is the logarithmic property that I can use to solve exponential equations?
A: The logarithmic property that you can use to solve exponential equations is $\log_a(b^c)=c\log_a(b)$. This property allows you to rewrite the exponential equation in a form that is easier to solve.
Q: How do I use the logarithmic property to solve an exponential equation?
A: To use the logarithmic property to solve an exponential equation, you need to take the logarithm of both sides of the equation. This will allow you to rewrite the equation in a form that is easier to solve.
Q: What is the next step after taking the logarithm of both sides of the equation?
A: After taking the logarithm of both sides of the equation, you need to isolate the variable on one side of the equation. You can do this by using algebraic properties, such as the property that states $\log_a(b)=c\log_a(a^c)$.
Q: How do I isolate the variable on one side of the equation?
A: To isolate the variable on one side of the equation, you need to use algebraic properties, such as the property that states $\log_a(b)=c\log_a(a^c)$. You can also use properties such as $\log_a(b^c)=c\log_a(b)$ and $\log_a(b)+\log_a(c)=\log_a(bc)$.
Q: What are some common mistakes to avoid when solving exponential equations?
A: Some common mistakes to avoid when solving exponential equations include:
- Not using the correct logarithmic property
- Not isolating the variable on one side of the equation
- Not checking the domain of the logarithmic function
- Not checking the range of the logarithmic function
Q: How do I check the domain and range of the logarithmic function?
A: To check the domain and range of the logarithmic function, you need to make sure that the base of the logarithm is positive and not equal to 1, and that the argument of the logarithm is positive.
Q: What are some real-world applications of exponential equations?
A: Exponential equations have many real-world applications, including:
- Modeling population growth
- Modeling chemical reactions
- Modeling financial investments
- Modeling electrical circuits
Conclusion
In this article, we have provided a Q&A guide on solving exponential equations. We have covered topics such as the definition of an exponential equation, the logarithmic property that can be used to solve exponential equations, and common mistakes to avoid when solving exponential equations. We hope that this guide has been helpful in providing a better understanding of exponential equations and how to solve them.
Final Answer
The final answer is:
B. $\frac{\log 7}{\log 5}$