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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving exponential equations of the form $a^{x-h} = b$, where $a$, $b$, and $h$ are constants. We will use the given equation $27^{x-3} = 81$ as a case study to demonstrate the step-by-step process of solving exponential equations.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and the base of the exponent is a constant. The general form of an exponential equation is $a^{x-h} = b$, where $a$ is the base, $x$ is the variable, $h$ is a constant, and $b$ is the constant on the right-hand side. In the given equation $27^{x-3} = 81$, $27$ is the base, $x-3$ is the exponent, and $81$ is the constant on the right-hand side.

Step 1: Identify the Base and the Constant

The first step in solving an exponential equation is to identify the base and the constant. In the given equation, the base is $27$, and the constant is $81$.

Step 2: Rewrite the Equation with the Same Base

To solve the equation, we need to rewrite it with the same base on both sides. Since $27$ and $81$ are both powers of $3$, we can rewrite the equation as follows:

$$27^{x-3} = (3^3)^{x-3} = 3^{3(x-3)} = 81$$

Step 3: Equate the Exponents

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

$$3(x-3) = 4$$

Step 4: Solve for x

To solve for $x$, we need to isolate the variable. We can do this by adding $3$ to both sides of the equation:

$$3x - 9 = 4$$

Next, we can add $9$ to both sides of the equation:

$$3x = 13$$

Finally, we can divide both sides of the equation by $3$ to solve for $x$:

$$x = \frac{13}{3}$$

Conclusion

Solving exponential equations requires a step-by-step approach. By identifying the base and the constant, rewriting the equation with the same base, equating the exponents, and solving for the variable, we can solve exponential equations of the form $a^{x-h} = b$. In this article, we used the equation $27^{x-3} = 81$ as a case study to demonstrate the step-by-step process of solving exponential equations.

Tips and Tricks

• Make sure to identify the base and the constant in the equation.
• Rewrite the equation with the same base on both sides.
• Equate the exponents since the bases are the same.
• Solve for the variable by isolating it on one side of the equation.

Common Mistakes to Avoid

• Failing to identify the base and the constant in the equation.
• Not rewriting the equation with the same base on both sides.
• Not equating the exponents since the bases are the same.
• Not solving for the variable by isolating it on one side of the equation.

Real-World Applications

Exponential equations have numerous real-world applications in fields such as finance, economics, and computer science. For example, exponential equations can be used to model population growth, compound interest, and the spread of diseases.

Conclusion

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Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves a variable in the exponent. The general form of an exponential equation is $a^{x-h} = b$, where $a$ is the base, $x$ is the variable, $h$ is a constant, and $b$ is the constant on the right-hand side.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to follow these steps:

1. Identify the base and the constant in the equation.
2. Rewrite the equation with the same base on both sides.
3. Equate the exponents since the bases are the same.
4. Solve for the variable by isolating it on one side of the equation.

Q: What if the bases are different?

A: If the bases are different, you need to rewrite the equation with the same base on both sides. For example, if you have the equation $2^x = 5^y$, you can rewrite it as $(2^x) = (5^y)$, and then use logarithms to solve for $x$ and $y$.

Q: How do I use logarithms to solve exponential equations?

A: To use logarithms to solve exponential equations, you need to take the logarithm of both sides of the equation. For example, if you have the equation $2^x = 5$, you can take the logarithm of both sides to get:

$$\log(2^x) = \log(5)$$

Using the property of logarithms that $\log(a^b) = b\log(a)$, you can rewrite the equation as:

$$x\log(2) = \log(5)$$

Finally, you can solve for $x$ by dividing both sides of the equation by $\log(2)$:

$$x = \frac{\log(5)}{\log(2)}$$

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Failing to identify the base and the constant in the equation.
• Not rewriting the equation with the same base on both sides.
• Not equating the exponents since the bases are the same.
• Not solving for the variable by isolating it on one side of the equation.

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug the solution back into the original equation and verify that it is true. For example, if you have the equation $2^x = 5$ and you solve for $x$ to get $x = \frac{\log(5)}{\log(2)}$, you can plug this value back into the original equation to get:

$$2^{\frac{\log(5)}{\log(2)}} = 5$$

Using the property of logarithms that $\log(a^b) = b\log(a)$, you can rewrite the equation as:

$$(2^{\frac{\log(5)}{\log(2)}})^{\log(2)} = 5^{\log(2)}$$

Finally, you can simplify the equation to get:

$$5 = 5$$

Since this is true, you can be confident that your solution is correct.

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

A: Exponential equations have numerous real-world applications in fields such as finance, economics, and computer science. For example, exponential equations can be used to model population growth, compound interest, and the spread of diseases.

Conclusion

Solving exponential equations is a crucial skill for students and professionals alike. By following the step-by-step process outlined in this article, you can solve exponential equations of the form $a^{x-h} = b$. Remember to identify the base and the constant, rewrite the equation with the same base, equate the exponents, and solve for the variable to ensure accurate solutions.