Solve For $x$:$6\left(1.12^x\right) = 17\left(1.03^x\right)$$ X = X = X = [/tex]

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic and exponential functions. In this article, we will focus on solving a specific type of exponential equation, which involves two exponential functions with different bases. We will use the given equation $6\left(1.12^x\right) = 17\left(1.03^x\right)$ as an example to demonstrate the step-by-step process of solving exponential equations.

Understanding Exponential Functions

Before we dive into solving the equation, let's take a moment to understand what exponential functions are. An exponential function is a function that can be written in the form $f(x) = a^x$, where $a$ is a positive constant 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 as $x$ increases.

The Given Equation

The given equation is $6\left(1.12^x\right) = 17\left(1.03^x\right)$. This equation involves two exponential functions with different bases: $1.12^x$ and $1.03^x$. Our goal is to solve for $x$.

Step 1: Isolate the Exponential Terms

To solve the equation, we need to isolate the exponential terms on one side of the equation. We can do this by dividing both sides of the equation by 6 and 17, respectively.

$$\frac{6\left(1.12^x\right)}{6} = \frac{17\left(1.03^x\right)}{17}$$

This simplifies to:

$$1.12^x = \frac{17}{6} \cdot 1.03^x$$

Step 2: Take the Natural Logarithm of Both Sides

To eliminate the exponential terms, we can take the natural logarithm of both sides of the equation. This will allow us to use the properties of logarithms to simplify the equation.

$$\ln\left(1.12^x\right) = \ln\left(\frac{17}{6} \cdot 1.03^x\right)$$

Using the property of logarithms that states $\ln(ab) = \ln(a) + \ln(b)$, we can rewrite the equation as:

$$x \ln(1.12) = \ln\left(\frac{17}{6}\right) + x \ln(1.03)$$

Step 3: Simplify the Equation

Now that we have isolated the exponential terms, we can simplify the equation by combining like terms.

$$x \ln(1.12) - x \ln(1.03) = \ln\left(\frac{17}{6}\right)$$

This simplifies to:

$$x \left(\ln(1.12) - \ln(1.03)\right) = \ln\left(\frac{17}{6}\right)$$

Step 4: Solve for $x$

Finally, we can solve for $x$ by dividing both sides of the equation by the coefficient of $x$.

$$x = \frac{\ln\left(\frac{17}{6}\right)}{\ln(1.12) - \ln(1.03)}$$

Conclusion

Solving exponential equations requires a deep understanding of algebraic and exponential functions. By following the step-by-step process outlined in this article, we can solve equations involving two exponential functions with different bases. The given equation $6\left(1.12^x\right) = 17\left(1.03^x\right)$ is a classic example of an exponential equation that can be solved using the natural logarithm. We hope that this article has provided a clear and concise guide to solving exponential equations.

Example Use Cases

Exponential equations have numerous applications in various fields, including:

• Finance: Exponential equations are used to model population growth, compound interest, and inflation.
• Biology: Exponential equations are used to model population growth, disease spread, and chemical reactions.
• Computer Science: Exponential equations are used to model algorithm complexity, data compression, and encryption.

Tips and Tricks

When solving exponential equations, it's essential to:

• Use the natural logarithm: The natural logarithm is a powerful tool for eliminating exponential terms.
• Simplify the equation: Combine like terms and simplify the equation to make it easier to solve.
• Check your work: Verify that your solution satisfies the original equation.

Conclusion

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function that can be written in the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term on one side of the equation and then use the natural logarithm to eliminate the exponential term. You can then solve for the variable $x$.

Q: What is the natural logarithm?

A: The natural logarithm is a mathematical function that is the inverse of the exponential function. It is denoted by $\ln(x)$ and is used to eliminate exponential terms in equations.

Q: How do I use the natural logarithm to solve an exponential equation?

A: To use the natural logarithm to solve an exponential equation, you need to take the natural logarithm of both sides of the equation. This will allow you to use the properties of logarithms to simplify the equation and solve for the variable $x$.

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

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

• Not isolating the exponential term: Make sure to isolate the exponential term on one side of the equation before using the natural logarithm.
• Not using the natural logarithm: The natural logarithm is a powerful tool for eliminating exponential terms. Make sure to use it when solving exponential equations.
• Not checking your work: Verify that your solution satisfies the original equation.

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

A: Exponential equations have numerous applications in various fields, including:

• Finance: Exponential equations are used to model population growth, compound interest, and inflation.
• Biology: Exponential equations are used to model population growth, disease spread, and chemical reactions.
• Computer Science: Exponential equations are used to model algorithm complexity, data compression, and encryption.

Q: How do I check my work when solving an exponential equation?

A: To check your work when solving an exponential equation, you need to verify that your solution satisfies the original equation. You can do this by plugging your solution back into the original equation and checking that it is true.

Q: What are some tips and tricks for solving exponential equations?

A: Some tips and tricks for solving exponential equations include:

• Use the natural logarithm: The natural logarithm is a powerful tool for eliminating exponential terms.
• Simplify the equation: Combine like terms and simplify the equation to make it easier to solve.
• Check your work: Verify that your solution satisfies the original equation.

Q: Can I use other logarithmic functions to solve exponential equations?

A: While the natural logarithm is the most commonly used logarithmic function for solving exponential equations, you can also use other logarithmic functions, such as the common logarithm or the base-10 logarithm. However, the natural logarithm is usually the most convenient and easiest to use.

Q: How do I graph exponential functions?

A: To graph exponential functions, you need to use a graphing calculator or a computer program. You can also use a table of values to create a graph of the function.

Q: What are some common exponential functions?

A: Some common exponential functions include:

• Exponential growth: $f(x) = a^x$, where $a$ is a positive constant.
• Exponential decay: $f(x) = a^{-x}$, where $a$ is a positive constant.
• Logarithmic functions: $f(x) = \log_a(x)$, where $a$ is a positive constant.

Conclusion

Solving exponential equations is a fundamental skill in mathematics that has numerous applications in various fields. By following the step-by-step process outlined in this article, you can solve equations involving exponential functions with different bases. We hope that this article has provided a clear and concise guide to solving exponential equations.