In The Following Equations, Solve For $x$ (to Two Decimal Places Where Necessary) And Write All The Steps Until The Final Answer.a. $5^{3x} - 5^{3x-1} = 4$b. $3^{x+1} \cdot 5 - 4 \cdot 3^{x+2} = -\frac{7}{3}$

Introduction

Exponential equations are a type of mathematical equation that involves an exponential function. In this article, we will focus on solving two exponential equations, $5^{3x} - 5^{3x-1} = 4$ and $3^{x+1} \cdot 5 - 4 \cdot 3^{x+2} = -\frac{7}{3}$. We will use algebraic techniques to isolate the variable $x$ and find its value to two decimal places where necessary.

Equation a: $5^{3x} - 5^{3x-1} = 4$

Step 1: Factor out the Common Term

The first step is to factor out the common term $5^{3x}$ from both terms on the left-hand side of the equation.

5^{3x} - 5^{3x-1} = 5^{3x}(1 - \frac{1}{5})

Step 2: Simplify the Equation

Now, we can simplify the equation by evaluating the expression inside the parentheses.

5^{3x}(1 - \frac{1}{5}) = 5^{3x}(\frac{4}{5})

Step 3: Divide Both Sides by the Common Term

Next, we can divide both sides of the equation by the common term $5^{3x}$.

\frac{5^{3x}(\frac{4}{5})}{5^{3x}} = \frac{4}{5}

Step 4: Simplify the Equation

Now, we can simplify the equation by canceling out the common term $5^{3x}$.

\frac{4}{5} = \frac{4}{5}

Step 5: Multiply Both Sides by 5

To get rid of the fraction, we can multiply both sides of the equation by 5.

5 \cdot \frac{4}{5} = 4

Step 6: Simplify the Equation

Now, we can simplify the equation by canceling out the common term.

4 = 4

Step 7: Take the Logarithm of Both Sides

To solve for $x$, we can take the logarithm of both sides of the equation.

\log(4) = \log(4)

Step 8: Use the Power Rule of Logarithms

Now, we can use the power rule of logarithms to rewrite the equation.

\log(4) = \log(4^{1})

Step 9: Simplify the Equation

Now, we can simplify the equation by evaluating the expression inside the logarithm.

\log(4) = 1 \cdot \log(4)

Step 10: Simplify the Equation

Now, we can simplify the equation by canceling out the common term.

\log(4) = \log(4)

Step 11: Use the Definition of Logarithm

Now, we can use the definition of logarithm to rewrite the equation.

\log(4) = x \cdot \log(5^{3})

Step 12: Simplify the Equation

Now, we can simplify the equation by evaluating the expression inside the logarithm.

\log(4) = x \cdot 3 \cdot \log(5)

Step 13: Divide Both Sides by 3

To isolate $x$, we can divide both sides of the equation by 3.

\frac{\log(4)}{3} = x \cdot \log(5)

Step 14: Divide Both Sides by $\log(5)$

To solve for $x$, we can divide both sides of the equation by $\log(5)$.

\frac{\log(4)}{3 \cdot \log(5)} = x

Step 15: Simplify the Equation

Now, we can simplify the equation by evaluating the expression inside the logarithm.

x = \frac{\log(4)}{3 \cdot \log(5)}

Step 16: Calculate the Value of $x$

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

x = \frac{\log(4)}{3 \cdot \log(5)} \approx 0.19

Equation b: $3^{x+1} \cdot 5 - 4 \cdot 3^{x+2} = -\frac{7}{3}$

Step 1: Factor out the Common Term

The first step is to factor out the common term $3^{x+1}$ from both terms on the left-hand side of the equation.

3^{x+1} \cdot 5 - 4 \cdot 3^{x+2} = 3^{x+1}(5 - 4 \cdot 3)

Step 2: Simplify the Equation

Now, we can simplify the equation by evaluating the expression inside the parentheses.

3^{x+1}(5 - 4 \cdot 3) = 3^{x+1}(-7)

Step 3: Divide Both Sides by $3^{x+1}$

To isolate the term with the variable, we can divide both sides of the equation by $3^{x+1}$.

\frac{3^{x+1}(-7)}{3^{x+1}} = -7

Step 4: Simplify the Equation

Now, we can simplify the equation by canceling out the common term $3^{x+1}$.

-7 = -7

Step 5: Add 7 to Both Sides

To get rid of the negative sign, we can add 7 to both sides of the equation.

-7 + 7 = 0

Step 6: Simplify the Equation

Now, we can simplify the equation by canceling out the common term.

0 = 0

Step 7: Take the Logarithm of Both Sides

To solve for $x$, we can take the logarithm of both sides of the equation.

\log(0) = \log(0)

Step 8: Use the Definition of Logarithm

Now, we can use the definition of logarithm to rewrite the equation.

\log(0) = x \cdot \log(3^{x+1})

Step 9: Simplify the Equation

Now, we can simplify the equation by evaluating the expression inside the logarithm.

\log(0) = x \cdot (x+1) \cdot \log(3)

Step 10: Divide Both Sides by $(x+1) \cdot \log(3)$

To isolate $x$, we can divide both sides of the equation by $(x+1) \cdot \log(3)$.

\frac{\log(0)}{(x+1) \cdot \log(3)} = x

Step 11: Simplify the Equation

Now, we can simplify the equation by evaluating the expression inside the logarithm.

x = \frac{\log(0)}{(x+1) \cdot \log(3)}

Step 12: Calculate the Value of $x$

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

x = \frac{\log(0)}{(x+1) \cdot \log(3)} \approx -0.01

Conclusion

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Q: What is an exponential equation?

A: An exponential equation is a type of mathematical equation that involves an exponential function. It is an equation in which the variable is raised to a power, and the power is also a variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use algebraic techniques such as factoring, simplifying, and isolating the variable. You can also use logarithms to solve exponential equations.

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

A: An exponential equation is an equation in which the variable is raised to a power, while a logarithmic equation is an equation in which the variable is the exponent of a power. For example, $2^x = 8$ is an exponential equation, while $\log_2(8) = x$ is a logarithmic equation.

Q: How do I use logarithms to solve an exponential equation?

A: To use logarithms to solve an exponential equation, you can take the logarithm of both sides of the equation. This will allow you to use the properties of logarithms to simplify the equation and isolate the variable.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_b(M^r) = r \cdot \log_b(M)$. This means that you can bring the exponent down in front of the logarithm.

Q: How do I use the power rule of logarithms to solve an exponential equation?

A: To use the power rule of logarithms to solve an exponential equation, you can take the logarithm of both sides of the equation and then use the power rule to bring the exponent down in front of the logarithm.

Q: What is the definition of a logarithm?

A: The definition of a logarithm is that $\log_b(M) = x$ means that $b^x = M$. This means that the logarithm is the inverse of the exponential function.

Q: How do I use the definition of a logarithm to solve an exponential equation?

A: To use the definition of a logarithm to solve an exponential equation, you can take the logarithm of both sides of the equation and then use the definition of a logarithm to rewrite the equation in exponential form.

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 correctly
• Not checking the domain of the logarithm
• Not using the correct method to solve the equation

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

A: To check your work when solving an exponential equation, you can:

• Plug your solution back into the original equation to make sure it is true
• Check that your solution is in the correct form (e.g. that it is a number or an expression)
• Check that your solution satisfies the original equation

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

A: Exponential equations have many real-world applications, including:

• Modeling population growth
• Modeling chemical reactions
• Modeling financial growth
• Modeling physical phenomena such as radioactive decay

Q: How do I use technology to solve exponential equations?

A: You can use technology such as calculators or computer software to solve exponential equations. These tools can help you to:

• Evaluate expressions
• Simplify equations
• Solve equations
• Graph functions

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

• Start by simplifying the equation
• Use the properties of logarithms to isolate the variable
• Check your work carefully
• Use technology to help you solve the equation

Conclusion

In this article, we have answered some frequently asked questions about solving exponential equations. We have covered topics such as the definition of an exponential equation, how to use logarithms to solve exponential equations, and how to check your work when solving exponential equations. We have also provided some tips for solving exponential equations and some real-world applications of exponential equations.