Solve The Equation.${ \log _8(3x + 1) + 16 = 18 }$ { x = [?] \}

Mar 11, 2025 by ADMIN 68 views

Solving the Equation: A Step-by-Step Guide

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Introduction

In this article, we will delve into the world of logarithmic equations and explore a step-by-step solution to the given equation. The equation in question is $\log _8(3x + 1) + 16 = 18$ . Our goal is to isolate the variable $x$ and find its value.

Understanding Logarithmic Equations

Before we dive into the solution, it's essential to understand the basics of logarithmic equations. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log _a(x)$ , then $a^y = x$ . This means that the logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number.

Step 1: Isolate the Logarithmic Term

The first step in solving the equation is to isolate the logarithmic term. We can do this by subtracting 16 from both sides of the equation:

\log _8(3x + 1) = 18 - 16

\log _8(3x + 1) = 2

Step 2: Rewrite the Equation in Exponential Form

Now that we have isolated the logarithmic term, we can rewrite the equation in exponential form. Since the base of the logarithm is 8, we can rewrite the equation as:

8^2 = 3x + 1

Step 3: Simplify the Equation

Next, we can simplify the equation by evaluating the exponent:

64 = 3x + 1

Step 4: Isolate the Variable

Now that we have simplified the equation, we can isolate the variable $x$ by subtracting 1 from both sides:

63 = 3x

Step 5: Solve for x

Finally, we can solve for $x$ by dividing both sides of the equation by 3:

x = \frac{63}{3}

x = 21

Conclusion

In this article, we have solved the equation $\log _8(3x + 1) + 16 = 18$ using a step-by-step approach. We isolated the logarithmic term, rewrote the equation in exponential form, simplified the equation, isolated the variable, and finally solved for $x$ . The value of $x$ is 21.

Frequently Asked Questions

Q: What is the base of the logarithm in the given equation? A: The base of the logarithm is 8.
Q: What is the value of $x$ in the given equation? A: The value of $x$ is 21.
Q: How do I solve a logarithmic equation? A: To solve a logarithmic equation, isolate the logarithmic term, rewrite the equation in exponential form, simplify the equation, isolate the variable, and finally solve for the variable.

Additional Resources

Logarithmic equations: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation.
Exponential form: Exponential form is a way of writing an equation that involves a base and an exponent.
Isolating the variable: Isolating the variable means getting the variable by itself on one side of the equation.

Final Thoughts

Solving logarithmic equations requires a step-by-step approach. By isolating the logarithmic term, rewriting the equation in exponential form, simplifying the equation, isolating the variable, and finally solving for the variable, we can find the value of the variable. In this article, we have solved the equation $\log _8(3x + 1) + 16 = 18$ and found the value of $x$ to be 21.

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Introduction

In our previous article, we solved the equation $\log _8(3x + 1) + 16 = 18$ using a step-by-step approach. However, we understand that logarithmic equations can be complex and may raise several questions. In this article, we will address some of the most frequently asked questions about logarithmic equations.

Q&A

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log _a(x)$ , then $a^y = x$ .

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, follow these steps:

Isolate the logarithmic term.
Rewrite the equation in exponential form.
Simplify the equation.
Isolate the variable.
Solve for the variable.

Q: What is the base of the logarithm in a logarithmic equation?

A: The base of the logarithm is the number that is raised to a power to produce the argument of the logarithm. For example, in the equation $\log _8(3x + 1)$ , the base of the logarithm is 8.

Q: How do I rewrite a logarithmic equation in exponential form?

A: To rewrite a logarithmic equation in exponential form, use the fact that $\log _a(x) = y$ is equivalent to $a^y = x$ . For example, the equation $\log _8(3x + 1) = 2$ can be rewritten as $8^2 = 3x + 1$ .

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

A: A logarithmic equation involves a logarithm, while an exponential equation involves an exponent. For example, the equation $\log _8(3x + 1) = 2$ is a logarithmic equation, while the equation $8^2 = 3x + 1$ is an exponential equation.

Q: Can I use a calculator to solve a logarithmic equation?

A: Yes, you can use a calculator to solve a logarithmic equation. However, it's essential to understand the concept behind the solution and to follow the steps outlined above.

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

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

Not isolating the logarithmic term.
Not rewriting the equation in exponential form.
Not simplifying the equation.
Not isolating the variable.
Not solving for the variable.

Conclusion

In this article, we have addressed some of the most frequently asked questions about logarithmic equations. We hope that this article has provided you with a better understanding of logarithmic equations and how to solve them. Remember to follow the steps outlined above and to avoid common mistakes.

Additional Resources

Logarithmic equations: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation.
Exponential form: Exponential form is a way of writing an equation that involves a base and an exponent.
Isolating the variable: Isolating the variable means getting the variable by itself on one side of the equation.

Final Thoughts

Solving logarithmic equations requires a step-by-step approach. By understanding the concept behind the solution and following the steps outlined above, you can solve even the most complex logarithmic equations. Remember to avoid common mistakes and to use a calculator when necessary.