Solve The Logarithmic Equation For T T T : 4 + 5 Log ⁡ 2 ( T + 4 ) = 44 4 + 5 \log_2(t + 4) = 44 4 + 5 Lo G 2 ​ ( T + 4 ) = 44 A. T = 8 T = 8 T = 8 B. T = 9 T = 9 T = 9 C. T = 256 T = 256 T = 256 D. T = 252 T = 252 T = 252

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 for the variable $t$. The equation is $4 + 5 \log_2(t + 4) = 44$. We will break down the solution step by step, using various mathematical techniques to arrive at the final answer.

Understanding Logarithmic Equations

Before we dive into the solution, let's briefly review what logarithmic equations are and how they work. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a(c) = b$. Logarithmic equations can be solved using various techniques, including algebraic manipulation, logarithmic properties, and numerical methods.

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 4 from both sides of the equation:

$$5 \log_2(t + 4) = 40$$

Step 2: Apply the Power Rule of Logarithms

The next step is to apply the power rule of logarithms, which states that $\log_a(M^b) = b \log_a(M)$. In this case, we can rewrite the equation as:

$$\log_2(t + 4) = 8$$

Step 3: Eliminate the Logarithm

Now that we have isolated the logarithmic term, we can eliminate it by exponentiating both sides of the equation. Since the base of the logarithm is 2, we can use the fact that $2^8 = 256$:

$$t + 4 = 256$$

Step 4: Solve for $t$

The final step is to solve for $t$ by subtracting 4 from both sides of the equation:

$$t = 252$$

Conclusion

In this article, we solved a logarithmic equation for the variable $t$. We used various mathematical techniques, including algebraic manipulation, logarithmic properties, and numerical methods, to arrive at the final answer. The solution involved isolating the logarithmic term, applying the power rule of logarithms, eliminating the logarithm, and solving for $t$. The final answer is $t = 252$.

Discussion

The solution to this logarithmic equation is not among the options provided. However, we can see that the correct answer is indeed $t = 252$, which is option D. This highlights the importance of carefully following the solution steps and using the correct mathematical techniques to arrive at the final answer.

Additional Tips and Tricks

When solving logarithmic equations, it's essential to remember the following tips and tricks:

• Always isolate the logarithmic term before applying any logarithmic properties.
• Use the power rule of logarithms to rewrite the equation in a more manageable form.
• Eliminate the logarithm by exponentiating both sides of the equation.
• Solve for the variable by performing the necessary algebraic manipulations.

By following these tips and tricks, you can become proficient in solving logarithmic equations and tackle even the most challenging problems with confidence.

Common Mistakes to Avoid

When solving logarithmic equations, it's easy to make mistakes. Here are some common mistakes to avoid:

• Failing to isolate the logarithmic term before applying logarithmic properties.
• Using the wrong logarithmic property or formula.
• Not eliminating the logarithm by exponentiating both sides of the equation.
• Not solving for the variable by performing the necessary algebraic manipulations.

By being aware of these common mistakes, you can avoid them and arrive at the correct solution.

Conclusion

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 $a^b = c$, then $\log_a(c) = b$.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term, apply the power rule of logarithms, eliminate the logarithm, and solve for the variable.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_a(M^b) = b \log_a(M)$. This means that you can rewrite a logarithmic expression with a power as a product of the logarithm and the power.

Q: How do I eliminate the logarithm?

A: To eliminate the logarithm, you need to exponentiate both sides of the equation. This means that you need to raise the base of the logarithm to the power of the exponent.

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, $2^3 = 8$ is an exponential equation, while $\log_2(8) = 3$ is a logarithmic 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 always a good idea to check your work by plugging the solution back into the original equation.

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

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

• Failing to isolate the logarithmic term before applying logarithmic properties.
• Using the wrong logarithmic property or formula.
• Not eliminating the logarithm by exponentiating both sides of the equation.
• Not solving for the variable by performing the necessary algebraic manipulations.

Q: How do I know if I have the correct solution to a logarithmic equation?

A: To check if you have the correct solution to a logarithmic equation, you need to plug the solution back into the original equation and verify that it is true.

Q: Can I use logarithmic equations in real-world applications?

A: Yes, 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: What are some tips for solving logarithmic equations?

A: Some tips for solving logarithmic equations include:

• Always isolate the logarithmic term before applying logarithmic properties.
• Use the power rule of logarithms to rewrite the equation in a more manageable form.
• Eliminate the logarithm by exponentiating both sides of the equation.
• Solve for the variable by performing the necessary algebraic manipulations.

Conclusion

Logarithmic equations can be challenging to solve, but with the right approach and techniques, they can be tackled with ease. By following the steps outlined in this article, you can become proficient in solving logarithmic equations and tackle even the most challenging problems with confidence. Remember to always isolate the logarithmic term, apply the power rule of logarithms, eliminate the logarithm, and solve for the variable. With practice and patience, you can master the art of solving logarithmic equations.