What Is The Solution Of $\log (2t + 4) = \log (14 - 3t)$?A. \[$-18\$\]B. \[$-2\$\]C. 2D. 10
Introduction
In this article, we will explore the solution of the given 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 provide us with the value of the variable t.
Understanding Logarithmic Equations
Before we dive into the solution, let's briefly discuss logarithmic equations. A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. The logarithmic function is denoted by log, and it is defined as the power to which a base number must be raised to produce a given value.
For example, if we have the equation $\log (x) = 2$, it means that the base number (x) raised to the power of 2 is equal to the given value. In this case, the base number is 10, and the given value is 100, so we can rewrite the equation as $10^2 = 100$.
Solving the Logarithmic Equation
Now, let's focus on solving the given logarithmic equation, $\log (2t + 4) = \log (14 - 3t)$. To solve this equation, we can use the property of logarithmic functions that states that if $\log (a) = \log (b)$, then a = b.
Using this property, we can rewrite the equation as:
Simplifying the Equation
Now, let's simplify the equation by combining like terms. We can start by adding 3t to both sides of the equation:
This simplifies to:
Isolating the Variable
Next, let's isolate the variable t by subtracting 4 from both sides of the equation:
This simplifies to:
Solving for t
Finally, let's solve for t by dividing both sides of the equation by 5:
This simplifies to:
Conclusion
In this article, we have explored the solution of the given logarithmic equation, $\log (2t + 4) = \log (14 - 3t)$. We have used the property of logarithmic functions that states that if $\log (a) = \log (b)$, then a = b, and we have simplified the equation to isolate the variable t. The solution to this equation is t = 2.
Final Answer
The final answer to the given logarithmic equation is t = 2.
Comparison with Options
Let's compare our solution with the given options:
A. B. C. 2 D. 10
Our solution, t = 2, matches option C.
Discussion
The solution to this logarithmic equation is t = 2. This means that when the value of t is 2, the equation $\log (2t + 4) = \log (14 - 3t)$ is satisfied.
Limitations
One limitation of this solution is that it assumes that the base number of the logarithmic function is 10. If the base number is different, the solution may be different.
Future Work
In the future, we can explore other logarithmic equations and their solutions. We can also investigate the properties of logarithmic functions and their applications in various fields.
References
- [1] "Logarithmic Functions" by Math Open Reference
- [2] "Logarithmic Equations" by Purplemath
- [3] "Solving Logarithmic Equations" by Mathway
Keywords
- Logarithmic equation
- Logarithmic function
- Exponential function
- Inverse function
- Algebra
- Calculus
- Mathematics
- Logarithmic properties
- Logarithmic applications
Introduction
In our previous article, we explored the solution of the logarithmic equation $\log (2t + 4) = \log (14 - 3t)$. In this article, we will answer some frequently asked questions related to logarithmic equations.
Q1: What is a logarithmic equation?
A1: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. The logarithmic function is denoted by log, and it is defined as the power to which a base number must be raised to produce a given value.
Q2: How do I solve a logarithmic equation?
A2: To solve a logarithmic equation, you can use the property of logarithmic functions that states that if $\log (a) = \log (b)$, then a = b. You can also use algebraic manipulations to isolate the variable.
Q3: What is the difference between a logarithmic equation and an exponential equation?
A3: A logarithmic equation involves a logarithmic function, while an exponential equation involves an exponential function. For example, the equation $\log (x) = 2$ is a logarithmic equation, while the equation $x^2 = 4$ is an exponential equation.
Q4: Can I use logarithmic equations to solve exponential equations?
A4: Yes, you can use logarithmic equations to solve exponential equations. By taking the logarithm of both sides of the exponential equation, you can convert it into a logarithmic equation.
Q5: What are some common properties of logarithmic functions?
A5: Some common properties of logarithmic functions include:
- The logarithm of 1 is 0: $\log (1) = 0$
- The logarithm of a number is the power to which the base number must be raised to produce that number: $\log (a) = b$ means $a = b^{\log (a)}$
- The logarithm of a product is the sum of the logarithms: $\log (ab) = \log (a) + \log (b)$
- The logarithm of a quotient is the difference of the logarithms: $\log (\frac{a}{b}) = \log (a) - \log (b)$
Q6: How do I use logarithmic properties to simplify an equation?
A6: You can use logarithmic properties to simplify an equation by applying the properties to the logarithmic terms. For example, if you have the equation $\log (ab) = \log (c)$, you can use the property $\log (ab) = \log (a) + \log (b)$ to rewrite the equation as $\log (a) + \log (b) = \log (c)$.
Q7: Can I use logarithmic equations to solve systems of equations?
A7: Yes, you can use logarithmic equations to solve systems of equations. By taking the logarithm of both sides of each equation, you can convert the system into a system of logarithmic equations.
Q8: What are some real-world applications of logarithmic equations?
A8: 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.
Q9: How do I choose the base number for a logarithmic equation?
A9: The choice of base number depends on the problem and the units of measurement. For example, if you are working with financial data, you may want to use a base number of 10, while if you are working with scientific data, you may want to use a base number of e.
Q10: Can I use logarithmic equations to solve trigonometric equations?
A10: Yes, you can use logarithmic equations to solve trigonometric equations. By taking the logarithm of both sides of the trigonometric equation, you can convert it into a logarithmic equation.
Conclusion
In this article, we have answered some frequently asked questions related to logarithmic equations. We have discussed the properties of logarithmic functions, how to solve logarithmic equations, and some real-world applications of logarithmic equations.
Final Answer
The final answer to the question "What is the solution of $\log (2t + 4) = \log (14 - 3t)$?" is t = 2.
Discussion
The solution to this logarithmic equation is t = 2. This means that when the value of t is 2, the equation $\log (2t + 4) = \log (14 - 3t)$ is satisfied.
Limitations
One limitation of this solution is that it assumes that the base number of the logarithmic function is 10. If the base number is different, the solution may be different.
Future Work
In the future, we can explore other logarithmic equations and their solutions. We can also investigate the properties of logarithmic functions and their applications in various fields.
References
- [1] "Logarithmic Functions" by Math Open Reference
- [2] "Logarithmic Equations" by Purplemath
- [3] "Solving Logarithmic Equations" by Mathway
Keywords
- Logarithmic equation
- Logarithmic function
- Exponential function
- Inverse function
- Algebra
- Calculus
- Mathematics
- Logarithmic properties
- Logarithmic applications