Which Equation Is Equivalent To $4^{x+3}=64$?A. $2^{x+6}=2^4$B. \$2^{2x+6}=2^6$[/tex\]C. $4^{2x+6}=4^2$D. $4^{x+3}=4^6$

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will explore the concept of exponential equations and provide a step-by-step guide on how to solve them. We will also examine a specific equation, $4^{x+3}=64$, and determine which of the given options is equivalent to it.

What are Exponential Equations?

Exponential equations are equations that involve an exponential expression, which is an expression of the form $a^x$, where $a$ is a positive real number and $x$ is a variable. Exponential equations can be written in the form $a^x=b$, where $a$ and $b$ are positive real numbers.

Properties of Exponential Equations

Exponential equations have several properties that make them useful in solving problems. Some of the key properties include:

• Exponentiation is commutative: $a^x=by$ implies $b^x=ay$.
• Exponentiation is associative: $(a^x)y=a^{xy}$.
• Exponentiation is distributive: $a^{x+y}=axa^y$.

Solving Exponential Equations

To solve an exponential equation, we need to isolate the variable $x$. We can do this by using the properties of exponentiation. Here are the steps to solve an exponential equation:

1. Write the equation in exponential form: Write the equation in the form $a^x=b$.
2. Use the properties of exponentiation: Use the properties of exponentiation to simplify the equation.
3. Isolate the variable: Isolate the variable $x$ by using the properties of exponentiation.

Solving the Equation $4^{x+3}=64$

Now, let's solve the equation $4^{x+3}=64$. We can start by writing the equation in exponential form:

$$4^{x+3}=64$$

We can rewrite $64$ as $4^3$:

$$4^{x+3}=4^3$$

Using the property of exponentiation that $a^{x+y}=axa^y$, we can rewrite the equation as:

$$4^x\cdot4^3=4^3$$

Now, we can divide both sides of the equation by $4^3$:

$$4^x=1$$

Using the property of exponentiation that $a^0=1$, we can rewrite the equation as:

$$x=0$$

Therefore, the solution to the equation $4^{x+3}=64$ is $x=0$.

Which Option is Equivalent to the Equation?

Now, let's examine the given options and determine which one is equivalent to the equation $4^{x+3}=64$.

Option A: $2^{x+6}=24$

We can rewrite $2^4$ as $16$:

$$2^{x+6}=16$$

Using the property of exponentiation that $a^{x+y}=axa^y$, we can rewrite the equation as:

$$2^x\cdot2^6=16$$

Now, we can divide both sides of the equation by $2^6$:

$$2^x=2^{-2}$$

Using the property of exponentiation that $a^{{-x}=\frac{1}{a}x}$, we can rewrite the equation as:

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

Simplifying the equation, we get:

$$2^x=\frac{1}{4}$$

Using the property of exponentiation that $a^x=b$ implies $x=\log_a b$, we can rewrite the equation as:

$$x=\log_2\frac{1}{4}$$

Simplifying the equation, we get:

$$x=-2$$

Therefore, option A is not equivalent to the equation $4^{x+3}=64$.

Option B: $2^{2x+6}=26$

We can rewrite $2^6$ as $64$:

$$2^{2x+6}=64$$

Using the property of exponentiation that $a^{x+y}=axa^y$, we can rewrite the equation as:

$$2^{2x}\cdot2^6=64$$

Now, we can divide both sides of the equation by $2^6$:

$$2^{2x}=2^{-2}$$

Using the property of exponentiation that $a^{{-x}=\frac{1}{a}x}$, we can rewrite the equation as:

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

Simplifying the equation, we get:

$$2^{2x}=\frac{1}{4}$$

Using the property of exponentiation that $a^x=b$ implies $x=\log_a b$, we can rewrite the equation as:

$$2x=\log_2\frac{1}{4}$$

Simplifying the equation, we get:

$$2x=-2$$

Dividing both sides of the equation by $2$, we get:

$$x=-1$$

Therefore, option B is not equivalent to the equation $4^{x+3}=64$.

Option C: $4^{2x+6}=42$

We can rewrite $4^2$ as $16$:

$$4^{2x+6}=16$$

Using the property of exponentiation that $a^{x+y}=axa^y$, we can rewrite the equation as:

$$4^{2x}\cdot4^6=16$$

Now, we can divide both sides of the equation by $4^6$:

$$4^{2x}=4^{-4}$$

Using the property of exponentiation that $a^{{-x}=\frac{1}{a}x}$, we can rewrite the equation as:

$$4^{2x}=\frac{1}{4^4}$$

Simplifying the equation, we get:

$$4^{2x}=\frac{1}{256}$$

Using the property of exponentiation that $a^x=b$ implies $x=\log_a b$, we can rewrite the equation as:

$$2x=\log_4\frac{1}{256}$$

Simplifying the equation, we get:

$$2x=-8$$

Dividing both sides of the equation by $2$, we get:

$$x=-4$$

Therefore, option C is not equivalent to the equation $4^{x+3}=64$.

Option D: $4^{x+3}=46$

We can rewrite $4^6$ as $4096$:

$$4^{x+3}=4096$$

Using the property of exponentiation that $a^{x+y}=axa^y$, we can rewrite the equation as:

$$4^x\cdot4^3=4096$$

Now, we can divide both sides of the equation by $4^3$:

$$4^x=1024$$

Using the property of exponentiation that $a^x=b$ implies $x=\log_a b$, we can rewrite the equation as:

$$x=\log_4 1024$$

Simplifying the equation, we get:

$$x=6$$

Therefore, option D is equivalent to the equation $4^{x+3}=64$.

Conclusion

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is an expression of the form $a^x$, where $a$ is a positive real number and $x$ is a variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable $x$. You can do this by using the properties of exponentiation, such as the commutative, associative, and distributive properties.

Q: What are some common properties of exponentiation?

A: Some common properties of exponentiation include:

• Exponentiation is commutative: $a^x=by$ implies $b^x=ay$.
• Exponentiation is associative: $(a^x)y=a^{xy}$.
• Exponentiation is distributive: $a^{x+y}=axa^y$.

Q: How do I use the properties of exponentiation to solve an exponential equation?

A: To use the properties of exponentiation to solve an exponential equation, you need to:

1. Write the equation in exponential form: Write the equation in the form $a^x=b$.
2. Use the properties of exponentiation: Use the properties of exponentiation to simplify the equation.
3. Isolate the variable: Isolate the variable $x$ by using the properties of exponentiation.

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

A: An exponential equation is an equation that involves an exponential expression, while a logarithmic equation is an equation that involves a logarithmic expression. A logarithmic expression is an expression of the form $\log_a b$, where $a$ is a positive real number and $b$ is a positive real number.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the variable $x$. You can do this by using the properties of logarithms, such as the product rule and the quotient rule.

Q: What are some common properties of logarithms?

A: Some common properties of logarithms include:

• Product rule: $\log_a (bc)=\log_a b + \log_a c$.
• Quotient rule: $\log_a \frac{b}{c}=\log_a b - \log_a c$.

Q: How do I use the properties of logarithms to solve a logarithmic equation?

A: To use the properties of logarithms to solve a logarithmic equation, you need to:

1. Write the equation in logarithmic form: Write the equation in the form $\log_a b=x$.
2. Use the properties of logarithms: Use the properties of logarithms to simplify the equation.
3. Isolate the variable: Isolate the variable $x$ by using the properties of logarithms.

Q: What is the relationship between exponential equations and logarithmic equations?

A: Exponential equations and logarithmic equations are related in that they are inverse operations. This means that if you have an exponential equation of the form $a^x=b$, you can convert it to a logarithmic equation of the form $\log_a b=x$, and vice versa.

Q: How do I convert an exponential equation to a logarithmic equation?

A: To convert an exponential equation to a logarithmic equation, you need to:

1. Write the equation in exponential form: Write the equation in the form $a^x=b$.
2. Use the definition of logarithms: Use the definition of logarithms to rewrite the equation in logarithmic form.

Q: How do I convert a logarithmic equation to an exponential equation?

A: To convert a logarithmic equation to an exponential equation, you need to:

1. Write the equation in logarithmic form: Write the equation in the form $\log_a b=x$.
2. Use the definition of exponentiation: Use the definition of exponentiation to rewrite the equation in exponential form.

Conclusion

In this article, we have answered some frequently asked questions about exponential equations and logarithmic equations. We have also discussed the properties of exponentiation and logarithms, and how to use them to solve equations. We hope that this article has been helpful in understanding these concepts.