Solve Log ⁡ 5 ( 8 − 3 X ) = Log ⁡ 5 20 \log_5(8-3x) = \log_5 20 Lo G 5 ​ ( 8 − 3 X ) = Lo G 5 ​ 20 For X X X .A) X = 14 X=14 X = 14 B) X = − 13 X=-13 X = − 13 C) X = − 8 X=-8 X = − 8 D) X = − 4 X=-4 X = − 4

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 5. We will break down the solution step by step, providing a clear understanding of the process.

The Given Equation

The equation we need to solve is:

$$\log_5(8-3x) = \log_5 20$$

Step 1: Understanding the Properties of Logarithms

Before we dive into solving the equation, it's essential to understand the properties of logarithms. One of the key properties is that if $\log_a b = \log_a c$, then $b = c$. This property will be crucial in solving the given equation.

Step 2: Equating the Arguments

Using the property mentioned above, we can equate the arguments of the logarithms:

$$8-3x = 20$$

Step 3: Solving for $x$

Now that we have a linear equation, we can solve for $x$:

$$8-3x = 20$$

$$-3x = 12$$

$$x = -4$$

Conclusion

In this article, we solved a logarithmic equation involving a base of 5. We used the properties of logarithms to equate the arguments and then solved for $x$. The final answer is $x = -4$.

Why Choose This Solution?

This solution is the correct answer because it satisfies the original equation. When we substitute $x = -4$ into the original equation, we get:

$$\log_5(8-3(-4)) = \log_5 20$$

$$\log_5(8+12) = \log_5 20$$

$$\log_5 20 = \log_5 20$$

This shows that the solution $x = -4$ is indeed correct.

Common Mistakes to Avoid

When solving logarithmic equations, it's essential to avoid common mistakes. One of the most common mistakes is to forget to equate the arguments of the logarithms. Another mistake is to not check the solution by substituting it back into the original equation.

Real-World Applications

Logarithmic equations have numerous real-world applications. For example, they are used in finance to calculate interest rates, in physics to describe the behavior of sound waves, and in computer science to model the growth of populations.

Final Thoughts

Solving logarithmic equations requires a clear understanding of the properties of logarithms and a step-by-step approach. By following the steps outlined in this article, you can solve logarithmic equations with ease. Remember to always check your solution by substituting it back into the original equation.

Additional Resources

For more information on logarithmic equations, check out the following resources:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

Conclusion

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. A logarithm is the inverse operation of exponentiation. In other words, it is the power to which a base must be raised to produce a given number.

Q: What are the common properties of logarithms?

A: There are several common properties of logarithms, including:

• Product Rule: $\log_a (b \cdot c) = \log_a b + \log_a c$
• Quotient Rule: $\log_a (b / c) = \log_a b - \log_a c$
• Power Rule: $\log_a (b^c) = c \cdot \log_a b$
• Change of Base Formula: $\log_b a = \frac{\log_c a}{\log_c b}$

Q: How do I solve a logarithmic equation?

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

1. Isolate the logarithm: Move all terms except the logarithm to one side of the equation.
2. Use the properties of logarithms: Use the product rule, quotient rule, power rule, or change of base formula to simplify the equation.
3. Exponentiate both sides: Raise both sides of the equation to the power of the base to eliminate the logarithm.
4. Solve for the variable: Solve for the variable in the resulting equation.

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:

• Logarithmic equation: $\log_2 x = 3$
• Exponential equation: $2^x = 8$

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

A: Yes, you can use a calculator to solve logarithmic equations. However, it's essential to understand the properties of logarithms and how to use them to simplify the equation before using a calculator.

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

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

• Forgetting to isolate the logarithm: Make sure to move all terms except the logarithm to one side of the equation.
• Not using the properties of logarithms: Use the product rule, quotient rule, power rule, or change of base formula to simplify the equation.
• Not exponentiating both sides: Raise both sides of the equation to the power of the base to eliminate the logarithm.
• Not checking the solution: Make sure to check the solution by substituting it back into the original equation.

Q: How do I check my solution to a logarithmic equation?

A: To check your solution to a logarithmic equation, substitute the solution back into the original equation and verify that it is true. For example:

• Original equation: $\log_2 x = 3$
• Solution: $x = 8$
• Verification: $\log_2 8 = 3$

Conclusion

In conclusion, logarithmic equations can be challenging to solve, but with the right approach and understanding of the properties of logarithms, you can solve them with ease. Remember to always check your solution by substituting it back into the original equation.

Additional Resources

For more information on logarithmic equations, check out the following resources:

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

Final Thoughts

Solving logarithmic equations requires a clear understanding of the properties of logarithms and a step-by-step approach. By following the steps outlined in this article and avoiding common mistakes, you can solve logarithmic equations with ease.