(10 Pts.) Solve For X X X . Give Both The Exact Solution And The Approximate Solution Rounded To Four Decimals: Log ⁡ 2 ( X + 1 ) − 2 Log ⁡ 2 ( X ) = 1 \log _2(x+1) - 2 \log _2(x) = 1 Lo G 2 ( X + 1 ) − 2 Lo G 2 ( X ) = 1

Mar 10, 2025 by ADMIN 226 views

**Solving Logarithmic Equations: A Step-by-Step Guide**

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 the equation $\log _2(x+1) - 2 \log _2(x) = 1$ . We will first solve the equation exactly and then find the approximate solution rounded to four decimals.

Understanding Logarithmic Properties

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In this case, we are dealing with base 2 logarithms.

One of the key properties of logarithms is the product rule, which states that $\log_b(xy) = \log_b(x) + \log_b(y)$ . Another important property is the power rule, which states that $\log_b(x^y) = y\log_b(x)$ .

Solving the Equation Exactly

To solve the equation $\log _2(x+1) - 2 \log _2(x) = 1$ , we can start by using the power rule to simplify the equation.

$\log _2(x+1) - 2 \log _2(x) = 1$

Using the power rule, we can rewrite the equation as:

$\log _2(x+1) - \log _2(x^2) = 1$

Now, we can use the quotient rule, which states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$ . Applying this rule, we get:

$\log _2(\frac{x+1}{x^2}) = 1$

Next, we can rewrite the equation in exponential form by raising the base (2) to both sides of the equation.

$2^1 = \frac{x+1}{x^2}$

Simplifying the equation, we get:

$2 = \frac{x+1}{x^2}$

Cross-multiplying, we get:

$2x^2 = x + 1$

Expanding the equation, we get:

$2x^2 - x - 1 = 0$

This is a quadratic equation, and we can solve it using the quadratic formula.

Solving the Quadratic Equation

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, $a = 2$ , $b = -1$ , and $c = -1$ . Plugging these values into the formula, we get:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-1)}}{2(2)}$

Simplifying the equation, we get:

$x = \frac{1 \pm \sqrt{1 + 8}}{4}$

$x = \frac{1 \pm \sqrt{9}}{4}$

$x = \frac{1 \pm 3}{4}$

This gives us two possible solutions:

$x = \frac{1 + 3}{4} = 1$

$x = \frac{1 - 3}{4} = -\frac{1}{2}$

However, we need to check if these solutions are valid. We can do this by plugging them back into the original equation.

Checking the Solutions

Plugging $x = 1$ into the original equation, we get:

$\log _2(1+1) - 2 \log _2(1) = 1$

$\log _2(2) - 2 \log _2(1) = 1$

$1 - 0 = 1$

This is true, so $x = 1$ is a valid solution.

Plugging $x = -\frac{1}{2}$ into the original equation, we get:

$\log _2(-\frac{1}{2}+1) - 2 \log _2(-\frac{1}{2}) = 1$

$\log _2(\frac{1}{2}) - 2 \log _2(-\frac{1}{2}) = 1$

$-1 - 2(-1) = 1$

$-1 + 2 = 1$

This is also true, so $x = -\frac{1}{2}$ is a valid solution.

Finding the Approximate Solution

To find the approximate solution, we can use a calculator or numerical methods. Let's use a calculator to find the approximate value of $x$ .

Using a calculator, we get:

$x \approx 1.0000$

This is the approximate value of $x$ rounded to four decimals.

Conclusion

Q: What is the main concept behind solving logarithmic equations?

A: The main concept behind solving logarithmic equations is to use logarithmic properties to simplify the equation and then solve for the variable.

Q: What are the key properties of logarithms that are used to solve logarithmic equations?

A: The key properties of logarithms that are used to solve logarithmic equations are the product rule, power rule, and quotient rule. These properties allow us to simplify the equation and isolate the variable.

Q: How do I use the product rule to simplify a logarithmic equation?

A: To use the product rule, you need to have a logarithm of a product. The product rule states that $\log_b(xy) = \log_b(x) + \log_b(y)$ . You can use this rule to break down the logarithm of a product into the sum of logarithms.

Q: How do I use the power rule to simplify a logarithmic equation?

A: To use the power rule, you need to have a logarithm of a power. The power rule states that $\log_b(x^y) = y\log_b(x)$ . You can use this rule to simplify the logarithm of a power by multiplying the exponent by the logarithm of the base.

Q: How do I use the quotient rule to simplify a logarithmic equation?

A: To use the quotient rule, you need to have a logarithm of a quotient. The quotient rule states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$ . You can use this rule to simplify the logarithm of a quotient by subtracting the logarithm of the denominator from the logarithm of the numerator.

Q: What is the quadratic formula, and how is it used to solve logarithmic equations?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The quadratic formula is used to solve logarithmic equations by first simplifying the equation to a quadratic equation and then using the quadratic formula to solve for the variable.

Q: How do I check if a solution is valid?

A: To check if a solution is valid, you need to plug the solution back into the original equation and check if it is true. If the solution satisfies the original equation, then it is a valid solution.

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

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

Not using the correct logarithmic properties
Not simplifying the equation correctly
Not checking if the solution is valid
Not using a calculator or numerical methods to find the approximate solution

Q: How do I find the approximate solution to a logarithmic equation?

A: To find the approximate solution to a logarithmic equation, you can use a calculator or numerical methods. You can also use software such as MATLAB or Python to find the approximate solution.

Q: What are some real-world applications of logarithmic equations?

A: 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.
Computer Science: Logarithmic equations are used to analyze and optimize algorithms.

Q: How can I practice solving logarithmic equations?

A: You can practice solving logarithmic equations by working through practice problems and exercises. You can also use online resources such as Khan Academy or MIT OpenCourseWare to practice solving logarithmic equations.