Solve The Following Equations:(9) ${ 5^x = \frac{1}{125} }$(10) ${ 7^{x-1} = \frac{1}{49} }$(11) ${ \left(\frac{1}{2}\right)^x = 2 }$(12) ${ \left(\frac{1}{2}\right)^x = 4 }$(13) $[
Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will explore four different exponential equations and provide step-by-step solutions to each of them.
Equation (9): 5^x = 1/125
To solve the equation 5^x = 1/125, we need to first simplify the right-hand side of the equation. We can rewrite 1/125 as 1/5^3, since 125 = 5^3.
5^x = 1/5^3
Using the properties of exponents, we can rewrite the equation as:
5^x = 5^(-3)
Since the bases are the same, we can equate the exponents:
x = -3
Therefore, the solution to the equation 5^x = 1/125 is x = -3.
Equation (10): 7^(x-1) = 1/49
To solve the equation 7^(x-1) = 1/49, we need to first simplify the right-hand side of the equation. We can rewrite 1/49 as 1/7^2, since 49 = 7^2.
7^(x-1) = 1/7^2
Using the properties of exponents, we can rewrite the equation as:
7^(x-1) = 7^(-2)
Since the bases are the same, we can equate the exponents:
x - 1 = -2
Solving for x, we get:
x = -1
Therefore, the solution to the equation 7^(x-1) = 1/49 is x = -1.
Equation (11): (1/2)^x = 2
To solve the equation (1/2)^x = 2, we need to first rewrite the right-hand side of the equation as a power of 2. We can rewrite 2 as 2^1.
(1/2)^x = 2^1
Using the properties of exponents, we can rewrite the equation as:
(1/2)^x = 2^1
Since the bases are the same, we can equate the exponents:
-x = 1
Solving for x, we get:
x = -1
Therefore, the solution to the equation (1/2)^x = 2 is x = -1.
Equation (12): (1/2)^x = 4
To solve the equation (1/2)^x = 4, we need to first rewrite the right-hand side of the equation as a power of 2. We can rewrite 4 as 2^2.
(1/2)^x = 2^2
Using the properties of exponents, we can rewrite the equation as:
(1/2)^x = 2^2
Since the bases are the same, we can equate the exponents:
-x = 2
Solving for x, we get:
x = -2
Therefore, the solution to the equation (1/2)^x = 4 is x = -2.
Conclusion
In this article, we have solved four different exponential equations using the properties of exponents. We have shown that by simplifying the right-hand side of the equation and using the properties of exponents, we can solve these equations and find the value of x. These techniques are essential in solving exponential equations and are widely used in mathematics and other fields.
Key Takeaways
- Exponential equations can be solved using the properties of exponents.
- Simplifying the right-hand side of the equation is essential in solving exponential equations.
- Using the properties of exponents, we can rewrite the equation and equate the exponents to solve for x.
- The solutions to the equations are x = -3, x = -1, x = -1, and x = -2.
Further Reading
For further reading on exponential equations, we recommend the following resources:
- Khan Academy: Exponential Equations
- Mathway: Exponential Equations
- Wolfram Alpha: Exponential Equations
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Calculus" by Michael Spivak
- [3] "Mathematics for the Nonmathematician" by Morris Kline
Frequently Asked Questions: Exponential Equations =====================================================
In this article, we will answer some of the most frequently asked questions about exponential equations. Whether you are a student, a teacher, or simply someone who wants to learn more about exponential equations, this article is for you.
Q: What is an exponential equation?
A: An exponential equation is an equation that involves an exponential expression, which is an expression of the form a^x, where a is a positive number and x is a variable.
Q: How do I solve an exponential equation?
A: To solve an exponential equation, you need to isolate the variable x. This can be done by using the properties of exponents, such as the product rule, the quotient rule, and the power rule.
Q: What is the product rule for exponents?
A: The product rule for exponents states that when you multiply two exponential expressions with the same base, you can add their exponents. For example, a^m * a^n = a^(m+n).
Q: What is the quotient rule for exponents?
A: The quotient rule for exponents states that when you divide two exponential expressions with the same base, you can subtract their exponents. For example, a^m / a^n = a^(m-n).
Q: What is the power rule for exponents?
A: The power rule for exponents states that when you raise an exponential expression to a power, you can multiply the exponents. For example, (am)n = a^(m*n).
Q: How do I simplify an exponential expression?
A: To simplify an exponential expression, you need to use the properties of exponents, such as the product rule, the quotient rule, and the power rule. You can also use the fact that a^0 = 1 and a^(-n) = 1/a^n.
Q: What is the difference between an exponential equation and a logarithmic equation?
A: An exponential equation is an equation that involves an exponential expression, while a logarithmic equation is an equation that involves a logarithmic expression. Logarithmic expressions are the inverse of exponential expressions.
Q: How do I solve a logarithmic equation?
A: To solve a logarithmic equation, you need to use the properties of logarithms, such as the product rule, the quotient rule, and the power rule. You can also use the fact that log(a) = x if and only if a^x = 1.
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 properties of exponents correctly
- Not simplifying the exponential expression before solving for x
- Not checking the domain of the exponential expression
- Not using the correct method to solve the equation
Q: How can I practice solving exponential equations?
A: You can practice solving exponential equations by working through examples and exercises in a textbook or online resource. You can also try solving real-world problems that involve exponential equations.
Q: What are some real-world applications of exponential equations?
A: Exponential equations have many real-world applications, including:
- Modeling population growth and decline
- Modeling chemical reactions
- Modeling financial growth and decline
- Modeling the spread of diseases
Conclusion
In this article, we have answered some of the most frequently asked questions about exponential equations. We have covered the basics of exponential equations, including how to solve them and how to simplify exponential expressions. We have also discussed some common mistakes to avoid and some real-world applications of exponential equations. Whether you are a student, a teacher, or simply someone who wants to learn more about exponential equations, this article is for you.
Key Takeaways
- Exponential equations are equations that involve an exponential expression.
- To solve an exponential equation, you need to isolate the variable x.
- The product rule, the quotient rule, and the power rule are essential tools for solving exponential equations.
- Simplifying the exponential expression before solving for x is crucial.
- Checking the domain of the exponential expression is essential.
- Real-world applications of exponential equations include modeling population growth and decline, modeling chemical reactions, and modeling financial growth and decline.
Further Reading
For further reading on exponential equations, we recommend the following resources:
- Khan Academy: Exponential Equations
- Mathway: Exponential Equations
- Wolfram Alpha: Exponential Equations
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Calculus" by Michael Spivak
- [3] "Mathematics for the Nonmathematician" by Morris Kline