Solve For \[$ X \$\] In The Equation:$\[ 6^{x-5} = 216^x \\]

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and their properties. In this article, we will focus on solving the equation $6^{x-5} = 216^x$ for the variable $x$. This equation involves exponential terms with different bases, and we will use various techniques to simplify and solve it.

Understanding Exponential Functions

Before we dive into solving the equation, let's briefly review the concept of exponential functions. An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable. The base $a$ determines the rate at which the function grows or decays. For example, the function $f(x) = 2^x$ grows rapidly as $x$ increases, while the function $f(x) = 0.5^x$ decays slowly.

Simplifying the Equation

To solve the equation $6^{x-5} = 216^x$, we can start by simplifying the right-hand side. We know that $216 = 6^3$, so we can rewrite the equation as:

$$6^{x-5} = (6^3)^x$$

Using the property of exponents that $(a^m)^n = a^{mn}$, we can simplify the right-hand side further:

$$6^{x-5} = 6^{3x}$$

Now we have two exponential terms with the same base, $6$. Since the bases are the same, we can equate the exponents:

$$x-5 = 3x$$

Solving for x

Now we have a linear equation in one variable, $x$. We can solve for $x$ by isolating the variable on one side of the equation. Subtracting $x$ from both sides gives us:

$$-5 = 2x$$

Dividing both sides by $2$ gives us:

$$x = -\frac{5}{2}$$

Verifying the Solution

To verify that our solution is correct, we can plug it back into the original equation:

$$6^{x-5} = 216^x$$

Substituting $x = -\frac{5}{2}$ into the equation gives us:

$$6^{-\frac{5}{2}-5} = 216^{-\frac{5}{2}}$$

Simplifying the exponents gives us:

$$6^{-\frac{15}{2}} = 6^{-15}$$

Since the bases are the same, we can equate the exponents:

$$-\frac{15}{2} = -15$$

This is a true statement, so we can conclude that our solution, $x = -\frac{5}{2}$, is correct.

Conclusion

Solving exponential equations requires a deep understanding of exponential functions and their properties. By simplifying the equation and equating the exponents, we can solve for the variable $x$. In this article, we solved the equation $6^{x-5} = 216^x$ for the variable $x$ and verified that our solution is correct. We hope that this article has provided a clear and concise guide to solving exponential equations.

Common Mistakes to Avoid

When solving exponential equations, there are several common mistakes to avoid. Here are a few:

• Not simplifying the equation: Failing to simplify the equation can make it difficult to solve. Make sure to simplify the equation as much as possible before solving for the variable.
• Not equating the exponents: When the bases are the same, it's essential to equate the exponents. Failing to do so can lead to incorrect solutions.
• Not verifying the solution: It's crucial to verify that the solution is correct by plugging it back into the original equation. This ensures that the solution is valid and not just a coincidence.

Real-World Applications

Exponential equations have numerous real-world applications. Here are a few examples:

• Population growth: Exponential equations can be used to model population growth. For example, the population of a city may grow exponentially over time, with the population doubling every few years.
• Financial modeling: Exponential equations can be used to model financial growth or decay. For example, an investment may grow exponentially over time, with the value increasing by a certain percentage each year.
• Science and engineering: Exponential equations are used extensively in science and engineering to model complex phenomena. For example, the decay of radioactive materials can be modeled using exponential equations.

Conclusion

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Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential term, which is a term of the form $a^x$, where $a$ is a positive real number and $x$ is the variable.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can start by rewriting the equation in a more manageable form. For example, if the equation involves a term with a large exponent, you can try to simplify the term by using the properties of exponents.

Q: What is the property of exponents that allows me to simplify an exponential equation?

A: The property of exponents that allows you to simplify an exponential equation is the rule that states $(a^m)^n = a^{mn}$. This rule allows you to simplify terms with large exponents by breaking them down into smaller terms.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can start by simplifying the equation and then equating the exponents. For example, if the equation is $a^x = b^y$, you can rewrite it as $a^x = (a^z)^y$, where $z$ is a constant. Then, you can equate the exponents and solve for $x$.

Q: What is the most common mistake to avoid when solving exponential equations?

A: The most common mistake to avoid when solving exponential equations is not simplifying the equation before solving for the variable. Failing to simplify the equation can make it difficult to solve and may lead to incorrect solutions.

Q: How do I verify that my solution to an exponential equation is correct?

A: To verify that your solution to an exponential equation is correct, you can plug it back into the original equation and check if it satisfies the equation. If the solution satisfies the equation, then it is correct.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have numerous real-world applications, including:

• Population growth: Exponential equations can be used to model population growth, where the population doubles every few years.
• Financial modeling: Exponential equations can be used to model financial growth or decay, where the value of an investment increases or decreases by a certain percentage each year.
• Science and engineering: Exponential equations are used extensively in science and engineering to model complex phenomena, such as the decay of radioactive materials.

Q: Can you provide an example of an exponential equation that is used in real-world applications?

A: Yes, here is an example of an exponential equation that is used in real-world applications:

$$P(t) = P_0 e^{rt}$$

This equation models population growth, where $P(t)$ is the population at time $t$, $P_0$ is the initial population, $r$ is the growth rate, and $t$ is time.

Q: How do I solve an exponential equation with a base that is not a power of 10?

A: To solve an exponential equation with a base that is not a power of 10, you can start by rewriting the equation in a more manageable form. For example, if the equation involves a term with a base of $e$, you can rewrite it as $e^x = e^y$, where $x$ and $y$ are the exponents.

Q: What is the difference between an exponential equation and a logarithmic equation?

A: An exponential equation is an equation that involves an exponential term, while a logarithmic equation is an equation that involves a logarithmic term. For example, the equation $a^x = b$ is an exponential equation, while the equation $\log_a b = x$ is a logarithmic equation.

Q: Can you provide an example of a logarithmic equation?

A: Yes, here is an example of a logarithmic equation:

$$\log_2 8 = x$$

This equation can be solved by rewriting it as $2^x = 8$, where $x$ is the exponent.

Conclusion

In conclusion, solving exponential equations requires a deep understanding of exponential functions and their properties. By simplifying the equation and equating the exponents, we can solve for the variable $x$. We hope that this article has provided a clear and concise guide to solving exponential equations. Remember to avoid common mistakes and verify your solution to ensure that it's correct.