If $\log_2(5x + 6) = 2$, Then $x = \square$You May Enter The Exact Value Or Round To 4 Decimal Places.

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific logarithmic equation involving a base of 2. The equation is $\log_2(5x + 6) = 2$, and we need to find the value of $x$. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's take a moment to understand what logarithmic equations are. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log_b(x)$, then $b^y = x$. This means that the logarithm of a number is the exponent to which the base must be raised to produce that number.

Step 1: Rewrite the Equation in Exponential Form

To solve the equation $\log_2(5x + 6) = 2$, we need to rewrite it in exponential form. This means that we need to get rid of the logarithm and express the equation in terms of exponents. We can do this by using the fact that if $y = \log_b(x)$, then $b^y = x$. In this case, we have:

$$\log_2(5x + 6) = 2$$

We can rewrite this equation as:

$$2^2 = 5x + 6$$

Step 2: Simplify the Equation

Now that we have rewritten the equation in exponential form, we can simplify it by evaluating the exponent. In this case, we have:

$$2^2 = 5x + 6$$

We know that $2^2 = 4$, so we can substitute this value into the equation:

$$4 = 5x + 6$$

Step 3: Isolate the Variable

Now that we have simplified the equation, we need to isolate the variable $x$. To do this, we can subtract 6 from both sides of the equation:

$$4 - 6 = 5x + 6 - 6$$

This simplifies to:

$$-2 = 5x$$

Step 4: Solve for $x$

Finally, we can solve for $x$ by dividing both sides of the equation by 5:

$$\frac{-2}{5} = \frac{5x}{5}$$

This simplifies to:

$$x = -\frac{2}{5}$$

Conclusion

In this article, we solved a logarithmic equation involving a base of 2. We broke down the solution into manageable steps, making it easy to follow and understand. We rewrote the equation in exponential form, simplified it, isolated the variable, and finally solved for $x$. The final answer is $x = -\frac{2}{5}$.

Additional Tips and Tricks

• When solving logarithmic equations, it's essential to remember that the base of the logarithm is the same as the base of the exponential function.
• To rewrite a logarithmic equation in exponential form, use the fact that if $y = \log_b(x)$, then $b^y = x$.
• When simplifying logarithmic equations, be careful not to confuse the base of the logarithm with the base of the exponential function.

Common Mistakes to Avoid

• When solving logarithmic equations, it's easy to get confused between the base of the logarithm and the base of the exponential function. Make sure to keep track of the base and use it correctly.
• When simplifying logarithmic equations, be careful not to introduce extraneous solutions. Make sure to check your work and verify that the solution satisfies the original equation.

Real-World Applications

Logarithmic equations have many real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design and optimize systems.

Conclusion

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Frequently Asked Questions

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 $y = \log_b(x)$, then $b^y = x$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to rewrite it in exponential form, simplify it, isolate the variable, and finally solve for $x$. Here's a step-by-step guide:

1. Rewrite the equation in exponential form.
2. Simplify the equation.
3. Isolate the variable.
4. Solve for $x$.

Q: What is the base of a logarithmic equation?

A: The base of a logarithmic equation is the number that is raised to a power to produce the result. For example, in the equation $\log_2(5x + 6) = 2$, the base is 2.

Q: How do I choose the base of a logarithmic equation?

A: The base of a logarithmic equation is usually given in the problem. If it's not given, you can choose any base that is convenient for you. However, it's essential to keep track of the base and use it correctly.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, the equation $\log_2(5x + 6) = 2$ is a logarithmic equation, while the equation $2^x = 5x + 6$ 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 not just rely on the calculator.

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

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

• Confusing the base of the logarithm with the base of the exponential function.
• Introducing extraneous solutions.
• Not checking the work and verifying that the solution satisfies the original equation.

Q: How do I apply logarithmic equations to real-world problems?

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

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design and optimize systems.

Q: Can I use logarithmic equations to solve problems involving negative numbers?

A: Yes, you can use logarithmic equations to solve problems involving negative numbers. However, it's essential to keep track of the base and use it correctly.

Q: What are some advanced topics in logarithmic equations?

A: Some advanced topics in logarithmic equations include:

• Logarithmic differentiation.
• Logarithmic integration.
• Logarithmic series.

Conclusion

In conclusion, logarithmic equations are a powerful tool for solving problems involving exponents and logarithms. By understanding the concept behind logarithmic equations and following the steps outlined in this article, you can become proficient in solving logarithmic equations and apply them to real-world problems.