Solve For $x$:$3^{-9x} = 11^{x-9}$Round Your Answer To The Nearest Thousandth. Do Not Round Any Intermediate Computations.$x =$

**Solving Exponential Equations: A Step-by-Step Guide** =====================================================

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will explore how to solve exponential equations, using the equation $3^{-9x} = 11^{x-9}$ as an example. We will break down the solution into manageable steps and provide a clear explanation of each step.

What are Exponential Equations?

Exponential equations are equations that involve exponential functions, which are functions of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. Exponential equations can be written in the form $a^x = b^y$, where $a$ and $b$ are positive constants and $x$ and $y$ are variables.

Step 1: Take the Logarithm of Both Sides

To solve the equation $3^{-9x} = 11^{x-9}$, we need to take the logarithm of both sides. This will allow us to use the properties of logarithms to simplify the equation.

\log(3^{-9x}) = \log(11^{x-9})

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

-9x\log(3) = (x-9)\log(11)

Step 2: Simplify the Equation

Now that we have taken the logarithm of both sides, we can simplify the equation by combining like terms.

-9x\log(3) = x\log(11) - 9\log(11)

Step 3: Isolate the Variable

To isolate the variable $x$, we need to get all the terms involving $x$ on one side of the equation.

-9x\log(3) - x\log(11) = -9\log(11)

Factoring out the common term $x$, we get:

x(-9\log(3) - \log(11)) = -9\log(11)

Step 4: Solve for $x$

Now that we have isolated the variable $x$, we can solve for $x$ by dividing both sides of the equation by the coefficient of $x$.

x = \frac{-9\log(11)}{-9\log(3) - \log(11)}

Rounding the Answer

Finally, we need to round our answer to the nearest thousandth.

x \approx \frac{-9\log(11)}{-9\log(3) - \log(11)} \approx 0.025

Conclusion

Solving exponential equations can be challenging, but with the right approach, they can be tackled with ease. By taking the logarithm of both sides, simplifying the equation, isolating the variable, and solving for $x$, we can find the solution to the equation $3^{-9x} = 11^{x-9}$. We hope this article has provided a clear and concise guide to solving exponential equations.

Frequently Asked Questions

Q: What is an exponential equation?

A: An exponential equation is an equation that involves exponential functions, which are functions of 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 take the logarithm of both sides, simplify the equation, isolate the variable, and solve for $x$.

Q: What is the logarithm of a number?

A: The logarithm of a number is the power to which a base number must be raised to produce that number. For example, the logarithm of 100 to the base 10 is 2, because $10^2 = 100$.

Q: How do I round a number to the nearest thousandth?

A: To round a number to the nearest thousandth, you need to look at the digit in the ten-thousandths place. If it is 5 or greater, you round up. If it is less than 5, you round down.

Q: What is the solution to the equation $3^{-9x} = 11^{x-9}$?

A: The solution to the equation $3^{-9x} = 11^{x-9}$ is approximately 0.025.