Solve For $x$:$2^x = 15^{(x-3)}$Round Your Answer To The Nearest Thousandth.\$x = \_\_\_$[/tex\]Type Your Numerical Answer Below.$\square$

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 $2^x = 15^{(x-3)}$ and provide a step-by-step guide on how to approach such problems.

Understanding Exponential Functions

Before we dive into solving the equation, let's briefly review exponential functions. An exponential function is a function of 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.

Properties of Exponential Functions

Exponential functions have several important properties that we will use to solve the equation. These properties include:

• Exponentiation: $a^x$ is the same as $(a^y)^z$ if $x = yz$
• Power Rule: $(a^x)^y = a^{xy}$
• Product Rule: $a^x \cdot a^y = a^{x+y}$
• Quotient Rule: $\frac{a^x}{a^y} = a^{x-y}$

Solving the Equation

Now that we have reviewed the properties of exponential functions, let's focus on solving the equation $2^x = 15^{(x-3)}$. To solve this equation, we will use the following steps:

Step 1: Take the Logarithm of Both Sides

The first step in solving the equation is to take the logarithm of both sides. This will allow us to use the properties of logarithms to simplify the equation.

$$\log(2^x) = \log(15^{(x-3)})$$

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

$$x\log(2) = (x-3)\log(15)$$

Step 2: Distribute the Logarithm

Next, we will distribute the logarithm on the right-hand side of the equation.

$$x\log(2) = x\log(15) - 3\log(15)$$

Step 3: Move the Terms with $x$ to One Side

Now, we will move the terms with $x$ to one side of the equation.

$$x\log(2) - x\log(15) = -3\log(15)$$

Step 4: Factor Out $x$

Next, we will factor out $x$ from the left-hand side of the equation.

$$x(\log(2) - \log(15)) = -3\log(15)$$

Step 5: Divide Both Sides by the Coefficient of $x$

Finally, we will divide both sides of the equation by the coefficient of $x$.

$$x = \frac{-3\log(15)}{\log(2) - \log(15)}$$

Step 6: Simplify the Expression

Using the property of logarithms that $\log(a) - \log(b) = \log(\frac{a}{b})$, we can simplify the expression.

$$x = \frac{-3\log(15)}{\log(\frac{2}{15})}$$

Step 7: Use a Calculator to Find the Value of $x$

To find the value of $x$, we will use a calculator to evaluate the expression.

$$x \approx \frac{-3(1.1761)}{-2.875}$$

$$x \approx 1.295$$

Conclusion

In this article, we have solved the equation $2^x = 15^{(x-3)}$ using the properties of exponential functions and logarithms. We have shown that the solution to the equation is $x \approx 1.295$. This result demonstrates the importance of understanding exponential functions and logarithms in solving mathematical problems.

Final Answer

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Introduction

In our previous article, we solved the equation $2^x = 15^{(x-3)}$ using the properties of exponential functions and logarithms. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used to solve exponential equations.

Q: What is an exponential equation?

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

Q: What are the properties of exponential functions?

A: Exponential functions have several important properties, including:

• Exponentiation: $a^x$ is the same as $(a^y)^z$ if $x = yz$
• Power Rule: $(a^x)^y = a^{xy}$
• Product Rule: $a^x \cdot a^y = a^{x+y}$
• Quotient Rule: $\frac{a^x}{a^y} = a^{x-y}$

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the following steps:

1. Take the logarithm of both sides of the equation.
2. Use the properties of logarithms to simplify the equation.
3. Isolate the variable $x$.
4. Use a calculator to find the value of $x$.

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

A: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. An exponential equation, on the other hand, involves an exponential function.

Q: Can I use a calculator to solve an exponential equation?

A: Yes, you can use a calculator to solve an exponential equation. However, it's always a good idea to understand the underlying concepts and techniques used to solve the equation.

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

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

• Not using the correct properties of logarithms.
• Not isolating the variable $x$ correctly.
• Not using a calculator to check the solution.

Q: Can I use exponential equations to model real-world problems?

A: Yes, exponential equations can be used to model real-world problems, such as population growth, chemical reactions, and financial investments.

Q: What are some examples of exponential equations in real-world problems?

A: Some examples of exponential equations in real-world problems include:

• Modeling population growth: $P(t) = P_0e^{kt}$, where $P(t)$ is the population at time $t$, $P_0$ is the initial population, $k$ is the growth rate, and $t$ is time.
• Modeling chemical reactions: $A(t) = A_0e^{-kt}$, where $A(t)$ is the amount of substance $A$ at time $t$, $A_0$ is the initial amount, $k$ is the decay rate, and $t$ is time.
• Modeling financial investments: $A(t) = A_0(1 + r)^t$, where $A(t)$ is the amount of money at time $t$, $A_0$ is the initial amount, $r$ is the interest rate, and $t$ is time.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts and techniques used to solve exponential equations. We have also discussed some common mistakes to avoid and provided examples of exponential equations in real-world problems. By understanding exponential equations and how to solve them, you can model and analyze real-world problems and make informed decisions.