Which Of The Following Shows The Equation $\log _4(x+6)=2$ Rewritten Using Logarithms?A. $4^{\log _4(x+6)}=2^2$ B. $ 4 Log 4 ( X + 6 ) = 4 4 4^{\log _4(x+6)}=4^4 4 L O G 4 ( X + 6 ) = 4 4 [/tex] C. $\log _4(x+6)=\log _4 16$ D. $\log _4(x+6)=\log
Understanding Logarithmic Equations
Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In this article, we will focus on rewriting logarithmic equations using logarithms.
The Given Equation
The given equation is $\log _4(x+6)=2$. This equation involves a logarithm with base 4, and we need to rewrite it using logarithms.
Rewriting the Equation
To rewrite the equation, we need to apply the property of logarithms that states $\log _a(x)=y \iff a^y=x$. This property allows us to rewrite the logarithmic equation in exponential form.
Option A: $4^{\log _4(x+6)}=2^2$
Option A involves rewriting the logarithmic equation in exponential form. We can rewrite the equation as $4^{\log _4(x+6)}=2^2$, which is equivalent to $x+6=16$. This is because $4^{\log _4(x+6)}$ is equal to $x+6$, and $2^2$ is equal to 4.
Option B: $4^{\log _4(x+6)}=4^4$
Option B also involves rewriting the logarithmic equation in exponential form. We can rewrite the equation as $4^{\log _4(x+6)}=4^4$, which is equivalent to $x+6=256$. This is because $4^{\log _4(x+6)}$ is equal to $x+6$, and $4^4$ is equal to 256.
Option C: $\log _4(x+6)=\log _4 16$
Option C involves rewriting the logarithmic equation in a different form. We can rewrite the equation as $\log _4(x+6)=\log _4 16$, which is equivalent to $x+6=16$. This is because the logarithm of a number is equal to the logarithm of another number if and only if the numbers are equal.
Option D: $\log _4(x+6)=\log _4 4^4$
Option D also involves rewriting the logarithmic equation in a different form. We can rewrite the equation as $\log _4(x+6)=\log _4 4^4$, which is equivalent to $x+6=256$. This is because the logarithm of a number is equal to the logarithm of another number if and only if the numbers are equal.
Conclusion
In conclusion, the correct answer is Option A: $4^{\log _4(x+6)}=2^2$. This is because it correctly rewrites the logarithmic equation in exponential form, which is equivalent to $x+6=16$. The other options are incorrect because they either rewrite the equation in a different form or do not correctly apply the property of logarithms.
Tips and Tricks
- When rewriting logarithmic equations, it is essential to apply the property of logarithms that states $\log _a(x)=y \iff a^y=x$.
- Make sure to correctly apply the property of logarithms to avoid errors.
- When rewriting logarithmic equations, it is helpful to use the change-of-base formula to rewrite the equation in a different form.
Common Mistakes
- One common mistake when rewriting logarithmic equations is to forget to apply the property of logarithms.
- Another common mistake is to incorrectly apply the property of logarithms, which can lead to errors in the rewritten equation.
Real-World Applications
Logarithmic equations have numerous real-world applications, including:
- Finance: Logarithmic equations are used to calculate interest rates and investment returns.
- Science: Logarithmic equations are used to model population growth and decay.
- Engineering: Logarithmic equations are used to design and optimize systems.
Conclusion
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, it is an equation that involves a logarithm with a certain base.
Q: What is the difference between a logarithmic equation and an exponential equation?
A: A logarithmic equation and an exponential equation are related but distinct concepts. An exponential equation involves an exponent, while a logarithmic equation involves a logarithm. For example, the equation $2^x=16$ is an exponential equation, while the equation $\log _2 16=x$ is a logarithmic equation.
Q: How do I rewrite a logarithmic equation using logarithms?
A: To rewrite a logarithmic equation using logarithms, you need to apply the property of logarithms that states $\log _a(x)=y \iff a^y=x$. This property allows you to rewrite the logarithmic equation in exponential form.
Q: What is the change-of-base formula?
A: The change-of-base formula is a formula that allows you to rewrite a logarithmic equation in a different form. It states that $\log _a(x)=\frac{\log _b(x)}{\log _b(a)}$, where $a$, $b$, and $x$ are positive real numbers.
Q: How do I use the change-of-base formula to rewrite a logarithmic equation?
A: To use the change-of-base formula, you need to identify the base of the logarithm and the argument of the logarithm. Then, you can rewrite the logarithmic equation using the change-of-base formula.
Q: What are some common mistakes to avoid when rewriting logarithmic equations?
A: Some common mistakes to avoid when rewriting logarithmic equations include:
- Forgetting to apply the property of logarithms
- Incorrectly applying the property of logarithms
- Not using the change-of-base formula when necessary
Q: How do I solve a logarithmic equation?
A: To solve a logarithmic equation, you need to isolate the variable. This can be done by applying the property of logarithms and using algebraic manipulations.
Q: What are some real-world applications of logarithmic equations?
A: Logarithmic equations have numerous real-world applications, including:
- Finance: Logarithmic equations are used to calculate interest rates and investment returns.
- Science: Logarithmic equations are used to model population growth and decay.
- Engineering: Logarithmic equations are used to design and optimize systems.
Q: How do I choose the correct base for a logarithmic equation?
A: The choice of base for a logarithmic equation depends on the problem and the context. In general, it is best to choose a base that is convenient and easy to work with.
Q: What are some common logarithmic equations?
A: Some common logarithmic equations include:
Q: How do I graph a logarithmic equation?
A: To graph a logarithmic equation, you need to use a graphing calculator or software. You can also use algebraic manipulations to rewrite the equation in a form that is easier to graph.
Conclusion
In conclusion, logarithmic equations are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding the properties of logarithms and how to rewrite logarithmic equations using logarithms, you can solve problems in various fields.