What Is The Solution Of Log ⁡ ( 2 T + 4 ) = Log ⁡ ( 14 − 3 T \log (2t + 4) = \log (14 - 3t Lo G ( 2 T + 4 ) = Lo G ( 14 − 3 T ]?A. -18 B. -2 C. 2 D. 10

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Introduction

In this article, we will explore the solution to the logarithmic equation $\log (2t + 4) = \log (14 - 3t)$. This equation involves logarithmic functions, which are a crucial concept in mathematics, particularly in algebra and calculus. The solution to this equation will be derived using algebraic manipulations and properties of logarithms.

Understanding Logarithmic Functions

Before we dive into the solution, it's essential to understand the concept of logarithmic functions. A logarithmic function is the inverse of an exponential function. In other words, if $y = a^x$, then $x = \log_a y$. The logarithmic function $\log_a x$ gives the power to which the base $a$ must be raised to obtain the number $x$.

Properties of Logarithmic Functions

There are several properties of logarithmic functions that we will use to solve the equation. These properties include:

• Product Rule: $\log_a (xy) = \log_a x + \log_a y$
• Quotient Rule: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$
• Power Rule: $\log_a x^y = y \log_a x$

Solving the Equation

Now that we have a good understanding of logarithmic functions and their properties, we can proceed to solve the equation.

$\log (2t + 4) = \log (14 - 3t)$

Using the Product Rule, we can rewrite the equation as:

$\log (2t + 4) = \log (14 - 3t)$

$\log (2t + 4) - \log (14 - 3t) = 0$

Using the Quotient Rule, we can rewrite the equation as:

$\log \left(\frac{2t + 4}{14 - 3t}\right) = 0$

Since the logarithm of a number is equal to zero if and only if the number is equal to 1, we can rewrite the equation as:

$\frac{2t + 4}{14 - 3t} = 1$

Cross-multiplying, we get:

$2t + 4 = 14 - 3t$

Simplifying the equation, we get:

$5t = 10$

Dividing both sides by 5, we get:

$t = 2$

Conclusion

In this article, we have solved the logarithmic equation $\log (2t + 4) = \log (14 - 3t)$. The solution to this equation is $t = 2$. This solution is obtained by using algebraic manipulations and properties of logarithmic functions.

Final Answer

The final answer to the equation is $\boxed{2}$.

Discussion

The solution to this equation is a crucial concept in mathematics, particularly in algebra and calculus. The properties of logarithmic functions, such as the product rule, quotient rule, and power rule, are essential in solving this equation. The solution to this equation can be applied to various real-world problems, such as finance, engineering, and science.

Related Topics

• Logarithmic functions
• Algebraic manipulations
• Properties of logarithmic functions
• Exponential functions
• Inverse functions

References

• [1] "Logarithmic Functions" by Khan Academy
• [2] "Algebraic Manipulations" by Math Open Reference
• [3] "Properties of Logarithmic Functions" by Wolfram MathWorld

Keywords

• Logarithmic functions
• Algebraic manipulations
• Properties of logarithmic functions
• Exponential functions
• Inverse functions
• Logarithmic equation
• Solution to logarithmic equation
• Algebra
• Calculus
• Mathematics
  

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Introduction

In our previous article, we solved the logarithmic equation $\log (2t + 4) = \log (14 - 3t)$ and obtained the solution $t = 2$. In this article, we will address some frequently asked questions (FAQs) about the solution to this equation.

Q&A

Q1: What is the significance of the solution $t = 2$?

A1: The solution $t = 2$ is significant because it satisfies the original equation $\log (2t + 4) = \log (14 - 3t)$. This means that when $t = 2$, the expressions $2t + 4$ and $14 - 3t$ are equal, and their logarithms are also equal.

Q2: How do I verify the solution $t = 2$?

A2: To verify the solution $t = 2$, you can substitute $t = 2$ into the original equation and check if it is true. If the equation holds true, then $t = 2$ is a valid solution.

Q3: What if I get a different solution for $t$?

A3: If you get a different solution for $t$, it may be due to a mistake in your calculations or a misunderstanding of the properties of logarithmic functions. Make sure to double-check your work and review the properties of logarithmic functions to ensure that you are using them correctly.

Q4: Can I use the solution $t = 2$ in real-world applications?

A4: Yes, the solution $t = 2$ can be used in real-world applications where the equation $\log (2t + 4) = \log (14 - 3t)$ is relevant. For example, in finance, the solution $t = 2$ can be used to determine the value of a investment or a loan.

Q5: How do I extend the solution to other logarithmic equations?

A5: To extend the solution to other logarithmic equations, you can use the properties of logarithmic functions, such as the product rule, quotient rule, and power rule. These properties can be used to simplify and solve more complex logarithmic equations.

Q6: What are some common mistakes to avoid when solving logarithmic equations?

A6: Some common mistakes to avoid when solving logarithmic equations include:

• Not using the correct properties of logarithmic functions
• Not simplifying the equation correctly
• Not checking the solution for validity
• Not considering the domain and range of the logarithmic function

Conclusion

In this article, we have addressed some frequently asked questions (FAQs) about the solution of $\log (2t + 4) = \log (14 - 3t)$. We have provided answers to questions about the significance of the solution, how to verify the solution, and how to extend the solution to other logarithmic equations. We have also highlighted some common mistakes to avoid when solving logarithmic equations.

Final Answer

The final answer to the FAQs is:

• Q1: The solution $t = 2$ is significant because it satisfies the original equation.
• Q2: To verify the solution $t = 2$, substitute $t = 2$ into the original equation and check if it is true.
• Q3: If you get a different solution for $t$, double-check your work and review the properties of logarithmic functions.
• Q4: Yes, the solution $t = 2$ can be used in real-world applications.
• Q5: To extend the solution to other logarithmic equations, use the properties of logarithmic functions.
• Q6: Some common mistakes to avoid when solving logarithmic equations include not using the correct properties of logarithmic functions, not simplifying the equation correctly, not checking the solution for validity, and not considering the domain and range of the logarithmic function.

Related Topics

• Logarithmic functions
• Algebraic manipulations
• Properties of logarithmic functions
• Exponential functions
• Inverse functions
• Logarithmic equation
• Solution to logarithmic equation
• Algebra
• Calculus
• Mathematics

References

• [1] "Logarithmic Functions" by Khan Academy
• [2] "Algebraic Manipulations" by Math Open Reference
• [3] "Properties of Logarithmic Functions" by Wolfram MathWorld

Keywords

• Logarithmic functions
• Algebraic manipulations
• Properties of logarithmic functions
• Exponential functions
• Inverse functions
• Logarithmic equation
• Solution to logarithmic equation
• Algebra
• Calculus
• Mathematics
• FAQs
• Frequently Asked Questions
• Solution to logarithmic equation
• Logarithmic functions
• Algebraic manipulations
• Properties of logarithmic functions
• Exponential functions
• Inverse functions