Solve Each Equation. Enter Your Responses And Explanations In The Space Provided. You May Also Use The Drawing Tool To Help Explain Or Support Your Answer.1. $10^n = 700$2. $3 \cdot 10^{2y} = 3600$3. $5 \cdot 2^x = \frac{5}{16}$

Exponential equations are a type of mathematical equation that involves an exponential expression. In this article, we will solve three exponential equations and provide explanations for each solution.

Equation 1: $10^n = 700$

To solve this equation, we need to find the value of $n$ that makes the equation true. We can start by rewriting the equation as:

$$10^n = 700$$

We can see that $10^2 = 100$ and $10^3 = 1000$, so we can try to find a value of $n$ that is between 2 and 3.

Step 1: Take the logarithm of both sides of the equation.

$$\log(10^n) = \log(700)$$

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

$$n \log(10) = \log(700)$$

Step 2: Simplify the equation.

Since $\log(10) = 1$, we can simplify the equation to:

$$n = \log(700)$$

Step 3: Evaluate the logarithm.

Using a calculator, we can find that $\log(700) \approx 2.8451$. Therefore, the value of $n$ that makes the equation true is:

$$n \approx 2.8451$$

However, since $n$ must be an integer, we can round the value to the nearest integer. In this case, we can round up to 3, since $10^3 = 1000$ is closer to 700 than $10^2 = 100$.

Answer: $n = 3$

Explanation: The value of $n$ that makes the equation true is 3, since $10^3 = 1000$ is closer to 700 than $10^2 = 100$.

Equation 2: $3 \cdot 10^{2y} = 3600$

To solve this equation, we need to find the value of $y$ that makes the equation true. We can start by rewriting the equation as:

$$3 \cdot 10^{2y} = 3600$$

We can see that $10^3 = 1000$ and $10^4 = 10000$, so we can try to find a value of $y$ that makes the equation true.

Step 1: Divide both sides of the equation by 3.

$$10^{2y} = \frac{3600}{3}$$

Step 2: Simplify the equation.

$$10^{2y} = 1200$$

Step 3: Take the logarithm of both sides of the equation.

$$\log(10^{2y}) = \log(1200)$$

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

$$2y \log(10) = \log(1200)$$

Step 4: Simplify the equation.

Since $\log(10) = 1$, we can simplify the equation to:

$$2y = \log(1200)$$

Step 5: Evaluate the logarithm.

Using a calculator, we can find that $\log(1200) \approx 3.0792$. Therefore, the value of $2y$ that makes the equation true is:

$$2y \approx 3.0792$$

Step 6: Divide both sides of the equation by 2.

$$y \approx \frac{3.0792}{2}$$

Answer: $y \approx 1.5396$

Explanation: The value of $y$ that makes the equation true is approximately 1.5396, since $10^{2y} = 1200$ is closer to 3600 than $10^{2y} = 1000$.

Equation 3: $5 \cdot 2^x = \frac{5}{16}$

To solve this equation, we need to find the value of $x$ that makes the equation true. We can start by rewriting the equation as:

$$5 \cdot 2^x = \frac{5}{16}$$

We can see that $2^4 = 16$, so we can try to find a value of $x$ that makes the equation true.

Step 1: Divide both sides of the equation by 5.

$$2^x = \frac{5}{16} \div 5$$

Step 2: Simplify the equation.

$$2^x = \frac{1}{16}$$

Step 3: Take the logarithm of both sides of the equation.

$$\log(2^x) = \log\left(\frac{1}{16}\right)$$

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

$$x \log(2) = \log\left(\frac{1}{16}\right)$$

Step 4: Simplify the equation.

Since $\log(2) \approx 0.3010$ and $\log\left(\frac{1}{16}\right) = -4 \log(2)$, we can simplify the equation to:

$$x \approx -4$$

Answer: $x \approx -4$

Explanation: The value of $x$ that makes the equation true is approximately -4, since $2^x = \frac{1}{16}$ is closer to $\frac{5}{16}$ than $2^x = 1$.

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In this article, we will answer some common questions about exponential equations and provide examples to help illustrate the concepts.

Q: What is an exponential equation?

A: An exponential equation is a type of mathematical equation that involves an exponential expression. It is an equation in which the variable is raised to a power, such as $2^x$ or $10^n$.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use logarithms to simplify the equation and find the value of the variable. You can take the logarithm of both sides of the equation and use the property of logarithms that $\log(a^b) = b \log(a)$ to rewrite the equation.

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

A: An exponential equation is an equation in which the variable is raised to a power, while a linear equation is an equation in which the variable is not raised to a power. For example, $2x + 3 = 5$ is a linear equation, while $2^x = 5$ is an exponential equation.

Q: Can I use logarithms to solve any type of equation?

A: No, logarithms can only be used to solve equations that involve exponential expressions. If the equation does not involve an exponential expression, you will need to use a different method to solve it.

Q: How do I know which base to use when solving an exponential equation?

A: The base of the exponential expression is usually the same as the base of the logarithm that you are using to solve the equation. For example, if the equation is $2^x = 5$, you would use a base-2 logarithm to solve it.

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

A: Yes, you can use a calculator to solve an exponential equation. However, you should always check your work to make sure that the solution is correct.

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

A: An exponential equation is an equation in which the variable is raised to a power, while a quadratic equation is an equation in which the variable is squared. For example, $x^2 + 3x + 2 = 0$ is a quadratic equation, while $2^x = 5$ is an exponential equation.

Q: Can I use algebraic methods to solve an exponential equation?

A: No, algebraic methods such as factoring and the quadratic formula cannot be used to solve exponential equations. You will need to use logarithms to solve exponential equations.

Q: How do I know if an equation is an exponential equation or a linear equation?

A: If the equation involves an exponential expression, such as $2^x$ or $10^n$, it is an exponential equation. If the equation does not involve an exponential expression, it is a linear equation.

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

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

Q: What are some common applications of exponential equations?

A: Exponential equations have many applications in fields such as physics, engineering, and economics. They can be used to model population growth, chemical reactions, and financial investments.

Q: Can I use exponential equations to solve problems in finance?

A: Yes, exponential equations can be used to solve problems in finance such as calculating compound interest and determining the value of investments.

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 base for the logarithm
• Not simplifying the equation correctly
• Not checking the solution to make sure it is correct

By following these tips and avoiding common mistakes, you can successfully solve exponential equations and apply them to real-world situations.