Solve For $x$. Log ⁡ 2 ( 3 X − 4 ) = 3 \log _2(3x-4)=3 Lo G 2 ​ ( 3 X − 4 ) = 3 X = X = X =

by ADMIN 94 views

Solve for xx: log⁡2(3x−4)=3\log_2(3x-4)=3

In this article, we will delve into the world of logarithms and solve for the variable xx in the given equation log⁡2(3x−4)=3\log_2(3x-4)=3. This equation involves a logarithmic function with base 2, and our goal is to isolate the variable xx and find its value.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's take a moment to understand what logarithmic equations are and how they work. A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. In other words, if y=2xy = 2^x, then x=log⁡2(y)x = \log_2(y).

The Given Equation

The given equation is log⁡2(3x−4)=3\log_2(3x-4)=3. This equation involves a logarithmic function with base 2, and our goal is to isolate the variable xx and find its value.

Step 1: Exponentiate Both Sides

To solve for xx, we need to get rid of the logarithm. We can do this by exponentiating both sides of the equation. Since the base of the logarithm is 2, we can use the fact that 2log⁡2(x)=x2^{\log_2(x)} = x.

\log_2(3x-4) = 3
2^{\log_2(3x-4)} = 2^3
3x-4 = 8

Step 2: Add 4 to Both Sides

Now that we have 3x−4=83x-4=8, we can add 4 to both sides of the equation to get rid of the negative term.

3x-4+4 = 8+4
3x = 12

Step 3: Divide Both Sides by 3

Finally, we can divide both sides of the equation by 3 to solve for xx.

\frac{3x}{3} = \frac{12}{3}
x = 4

In this article, we solved for the variable xx in the given equation log⁡2(3x−4)=3\log_2(3x-4)=3. We used the properties of logarithmic functions to exponentiate both sides of the equation and then solved for xx by adding 4 to both sides and dividing both sides by 3. The final answer is x=4x = 4.

Example Use Cases

This equation has many real-world applications, such as:

  • Computer Science: Logarithmic equations are used in computer science to solve problems involving large datasets and complex algorithms.
  • Engineering: Logarithmic equations are used in engineering to solve problems involving electrical circuits and signal processing.
  • Finance: Logarithmic equations are used in finance to solve problems involving investment returns and risk analysis.

Tips and Tricks

Here are some tips and tricks to help you solve logarithmic equations:

  • Use the properties of logarithmic functions: Logarithmic functions have many properties that can be used to simplify and solve equations.
  • Exponentiate both sides: Exponentiating both sides of the equation can help you get rid of the logarithm and solve for the variable.
  • Add and subtract terms: Adding and subtracting terms can help you isolate the variable and solve for its value.

Common Mistakes

Here are some common mistakes to avoid when solving logarithmic equations:

  • Forgetting to exponentiate both sides: Failing to exponentiate both sides of the equation can lead to incorrect solutions.
  • Not using the properties of logarithmic functions: Failing to use the properties of logarithmic functions can make it difficult to solve the equation.
  • Not checking the domain: Failing to check the domain of the logarithmic function can lead to incorrect solutions.
    Solve for xx: log⁡2(3x−4)=3\log_2(3x-4)=3 - Q&A

In our previous article, we solved for the variable xx in the given equation log⁡2(3x−4)=3\log_2(3x-4)=3. In this article, we will answer some frequently asked questions about logarithmic equations and provide additional tips and tricks to help you solve them.

Q: What is the difference between a logarithmic equation and an exponential equation?

A: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. An exponential equation, on the other hand, is an equation that involves an exponential function. For example, log⁡2(3x−4)=3\log_2(3x-4)=3 is a logarithmic equation, while 2x=82^x=8 is an exponential equation.

Q: How do I know which base to use when solving a logarithmic equation?

A: The base of the logarithm is usually given in the problem. If it's not given, you can use the fact that log⁡a(x)=log⁡b(x)log⁡b(a)\log_a(x) = \frac{\log_b(x)}{\log_b(a)} to change the base to a more convenient one.

Q: Can I use the same properties of logarithmic functions to solve exponential equations?

A: No, the properties of logarithmic functions are not the same as the properties of exponential functions. While logarithmic functions have properties like log⁡a(xy)=log⁡a(x)+log⁡a(y)\log_a(xy) = \log_a(x) + \log_a(y), exponential functions have properties like ax+y=ax⋅aya^{x+y} = a^x \cdot a^y.

Q: How do I check the domain of a logarithmic function?

A: To check the domain of a logarithmic function, you need to make sure that the argument of the logarithm is positive. For example, in the equation log⁡2(3x−4)=3\log_2(3x-4)=3, the argument of the logarithm is 3x−43x-4, which must be positive. Therefore, we need to solve the inequality 3x−4>03x-4>0 to find the domain of the function.

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

A: Yes, you can use a calculator to solve logarithmic equations. However, you need to make sure that the calculator is set to the correct base and that you are using the correct function. For example, to solve the equation log⁡2(3x−4)=3\log_2(3x-4)=3 using a calculator, you would need to set the calculator to base 2 and use the log⁡\log function.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function. For example, to graph the function y=log⁡2(x)y = \log_2(x), you would need to create a table of values with xx values ranging from 1 to 10 and corresponding yy values.

Q: Can I use logarithmic equations to model real-world problems?

A: Yes, logarithmic equations can be used to model real-world problems. For example, the equation log⁡2(3x−4)=3\log_2(3x-4)=3 can be used to model the growth of a population or the decay of a substance.

Q: How do I choose the correct base for a logarithmic equation?

A: The choice of base depends on the problem and the units of measurement. For example, if you are working with a problem involving time, you may want to use a base of 10, while if you are working with a problem involving distance, you may want to use a base of 2.

Q: Can I use logarithmic equations to solve systems of equations?

A: Yes, logarithmic equations can be used to solve systems of equations. For example, you can use the equation log⁡2(3x−4)=3\log_2(3x-4)=3 to solve a system of equations involving two variables.

In this article, we answered some frequently asked questions about logarithmic equations and provided additional tips and tricks to help you solve them. We also discussed how to graph a logarithmic function and how to use logarithmic equations to model real-world problems.