Solve The Equation: Log ⁡ 2 ( X ) − 7 = Log ⁡ 2 3 \log _2(x) - 7 = \log _2 3 Lo G 2 ​ ( X ) − 7 = Lo G 2 ​ 3

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Introduction

Logarithmic equations can be challenging to solve, especially when they involve different bases. In this article, we will focus on solving the equation log⁡2(x)−7=log⁡23\log _2(x) - 7 = \log _2 3. We will break down the solution into manageable steps, using properties of logarithms to simplify the equation.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm. The logarithm of a number is the power to which a base number must be raised to produce that number. For example, log⁡28=3\log _2 8 = 3 because 23=82^3 = 8. Logarithmic equations can be solved using properties of logarithms, such as the product rule, quotient rule, and power rule.

The Equation

The equation we are solving is log⁡2(x)−7=log⁡23\log _2(x) - 7 = \log _2 3. Our goal is to isolate the variable xx and find its value.

Step 1: Add 7 to Both Sides

To start solving the equation, we add 7 to both sides:

log⁡2(x)−7+7=log⁡23+7\log _2(x) - 7 + 7 = \log _2 3 + 7

This simplifies to:

log⁡2(x)=log⁡23+7\log _2(x) = \log _2 3 + 7

Step 2: Use the Product Rule

The product rule states that log⁡a(xy)=log⁡ax+log⁡ay\log _a (xy) = \log _a x + \log _a y. We can use this rule to rewrite the right-hand side of the equation:

log⁡2(x)=log⁡2(3⋅27)\log _2(x) = \log _2 (3 \cdot 2^7)

Step 3: Simplify the Right-Hand Side

We can simplify the right-hand side by evaluating the expression inside the logarithm:

log⁡2(x)=log⁡2(3⋅128)\log _2(x) = \log _2 (3 \cdot 128)

log⁡2(x)=log⁡2(384)\log _2(x) = \log _2 (384)

Step 4: Use the Definition of Logarithm

The definition of logarithm states that log⁡ax=y\log _a x = y if and only if ay=xa^y = x. We can use this definition to rewrite the equation:

2log⁡2(x)=2log⁡2(384)2^{\log _2(x)} = 2^{\log _2 (384)}

Step 5: Simplify the Equation

We can simplify the equation by canceling out the logarithms:

x=384x = 384

Conclusion

We have solved the equation log⁡2(x)−7=log⁡23\log _2(x) - 7 = \log _2 3 by using properties of logarithms. We added 7 to both sides, used the product rule, simplified the right-hand side, and finally used the definition of logarithm to find the value of xx. The final answer is x=384x = 384.

Tips and Tricks

  • When solving logarithmic equations, it's essential to use properties of logarithms to simplify the equation.
  • Always check your work by plugging the solution back into the original equation.
  • If you're stuck, try using a different base or simplifying the equation using algebraic manipulations.

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 chemical reactions.
  • Engineering: Logarithmic equations are used to design electronic circuits and optimize system performance.

Common Mistakes

  • Forgetting to use properties of logarithms: When solving logarithmic equations, it's essential to use properties of logarithms to simplify the equation.
  • Not checking work: Always check your work by plugging the solution back into the original equation.
  • Using the wrong base: Make sure to use the correct base when solving logarithmic equations.

Conclusion

Solving logarithmic equations requires a deep understanding of properties of logarithms and algebraic manipulations. By following the steps outlined in this article, you can solve even the most challenging logarithmic equations. Remember to always check your work and use properties of logarithms to simplify the equation. With practice and patience, you'll become a master of solving logarithmic equations.

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Introduction

Logarithmic equations can be challenging to solve, especially for those who are new to the concept. In this article, we will answer some of the most frequently asked questions about logarithmic equations, providing step-by-step explanations and examples to help you understand the concepts.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. The logarithm of a number is the power to which a base number must be raised to produce that number. For example, log⁡28=3\log _2 8 = 3 because 23=82^3 = 8.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to use properties of logarithms to simplify the equation. You can use the product rule, quotient rule, and power rule to rewrite the equation in a more manageable form. Then, you can use algebraic manipulations to isolate the variable.

Q: What is the product rule for logarithms?

A: The product rule for logarithms states that log⁡a(xy)=log⁡ax+log⁡ay\log _a (xy) = \log _a x + \log _a y. This means that you can rewrite a logarithmic expression as the sum of two logarithmic expressions.

Q: What is the quotient rule for logarithms?

A: The quotient rule for logarithms states that log⁡axy=log⁡ax−log⁡ay\log _a \frac{x}{y} = \log _a x - \log _a y. This means that you can rewrite a logarithmic expression as the difference of two logarithmic expressions.

Q: What is the power rule for logarithms?

A: The power rule for logarithms states that log⁡axy=ylog⁡ax\log _a x^y = y \log _a x. This means that you can rewrite a logarithmic expression as the product of two logarithmic expressions.

Q: How do I use the definition of logarithm to solve an equation?

A: The definition of logarithm states that log⁡ax=y\log _a x = y if and only if ay=xa^y = x. You can use this definition to rewrite a logarithmic equation in a more manageable form.

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 exponential expression. For example, log⁡2(x)−7=log⁡23\log _2(x) - 7 = \log _2 3 is a logarithmic equation, while 2x=82^x = 8 is an exponential equation.

Q: How do I check my work when solving a logarithmic equation?

A: To check your work, you need to plug the solution back into the original equation and verify that it is true. This will help you ensure that your solution is correct.

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 use properties of logarithms
  • Not checking work
  • Using the wrong base
  • Not simplifying the equation enough

Conclusion

Logarithmic equations can be challenging to solve, but with practice and patience, you can become proficient in solving them. By following the steps outlined in this article and using properties of logarithms, you can solve even the most challenging logarithmic equations. Remember to always check your work and use properties of logarithms to simplify the equation.

Additional Resources

  • Logarithmic Equations Tutorial: A step-by-step tutorial on solving logarithmic equations.
  • Logarithmic Equations Practice Problems: A set of practice problems to help you improve your skills in solving logarithmic equations.
  • Logarithmic Equations Calculator: A calculator that can help you solve logarithmic equations.

Final Tips

  • Practice, practice, practice: The more you practice solving logarithmic equations, the more comfortable you will become with the concepts.
  • Use properties of logarithms: Properties of logarithms are essential in solving logarithmic equations. Make sure to use them to simplify the equation.
  • Check your work: Always check your work by plugging the solution back into the original equation.