Solve For $x$: (Write The Answer As A Reduced Fraction.) Log ⁡ 4 5 − Log ⁡ 4 X = 2 \log _4 5 - \log _4 X = 2 Lo G 4 5 − Lo G 4 X = 2 X = X = X = Enter Your Answer.

Mar 10, 2025 by ADMIN 170 views

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

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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 type of logarithmic equation, namely the equation involving logarithms with the same base. We will use the given equation $\log _4 5 - \log _4 x = 2$ as an example to demonstrate the steps involved in solving such equations.

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 logarithm, which is the inverse operation of exponentiation. In other words, if $a^b = c$ , then $\log_a c = b$ . Logarithmic equations can be solved using various techniques, including the use of logarithmic properties and the change of base formula.

The Given Equation

The given equation is $\log _4 5 - \log _4 x = 2$ . This equation involves two logarithms with the same base, namely 4. We can use the properties of logarithms to simplify this equation and solve for $x$ .

Using Logarithmic Properties

One of the key properties of logarithms is the product rule, which states that $\log_a (m \cdot n) = \log_a m + \log_a n$ . We can use this property to rewrite the given equation as follows:

$\log _4 5 - \log _4 x = 2$

$\log _4 \left(\frac{5}{x}\right) = 2$

This step involves using the quotient rule, which states that $\log_a \left(\frac{m}{n}\right) = \log_a m - \log_a n$ . By applying this rule, we can rewrite the original equation as a single logarithm.

Applying the Change of Base Formula

The change of base formula is another important property of logarithms, which states that $\log_a b = \frac{\log_c b}{\log_c a}$ . We can use this formula to rewrite the equation in terms of a common logarithm, such as the natural logarithm or the logarithm with base 10.

$\log _4 \left(\frac{5}{x}\right) = 2$

$\frac{\log \left(\frac{5}{x}\right)}{\log 4} = 2$

This step involves using the change of base formula to rewrite the equation in terms of a common logarithm.

Solving for $x$

Now that we have rewritten the equation in a more manageable form, we can solve for $x$ . To do this, we can start by isolating the logarithm on one side of the equation.

$\frac{\log \left(\frac{5}{x}\right)}{\log 4} = 2$

$\log \left(\frac{5}{x}\right) = 2 \cdot \log 4$

$\log \left(\frac{5}{x}\right) = \log 4^2$

$\frac{5}{x} = 4^2$

$\frac{5}{x} = 16$

$x = \frac{5}{16}$

This is the solution to the given equation. We can verify this solution by plugging it back into the original equation.

Conclusion

Solving logarithmic equations can be challenging, but with the right approach, they can be tackled with ease. In this article, we used the given equation $\log _4 5 - \log _4 x = 2$ as an example to demonstrate the steps involved in solving such equations. We used the properties of logarithms, including the product rule and the change of base formula, to simplify the equation and solve for $x$ . The solution to the equation is $x = \frac{5}{16}$ .

Final Answer

The final answer is: $\boxed{\frac{5}{16}}$

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Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $a^b = c$ , then $\log_a c = b$ . Logarithmic equations can be solved using various techniques, including the use of logarithmic properties and the change of base formula.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the following steps:

Use the properties of logarithms to simplify the equation.
Apply the change of base formula to rewrite the equation in terms of a common logarithm.
Isolate the logarithm on one side of the equation.
Use the definition of a logarithm to rewrite the equation in exponential form.
Solve for the variable.

Q: What is the product rule for logarithms?

A: The product rule for logarithms states that $\log_a (m \cdot n) = \log_a m + \log_a n$ . This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: What is the quotient rule for logarithms?

A: The quotient rule for logarithms states that $\log_a \left(\frac{m}{n}\right) = \log_a m - \log_a n$ . This means that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q: What is the change of base formula?

A: The change of base formula is a property of logarithms that states that $\log_a b = \frac{\log_c b}{\log_c a}$ . This means that the logarithm of a number with a given base can be rewritten in terms of a common logarithm.

Q: How do I apply the change of base formula?

A: To apply the change of base formula, you can follow these steps:

Identify the base of the logarithm in the equation.
Choose a common logarithm, such as the natural logarithm or the logarithm with base 10.
Rewrite the equation using the change of base formula.
Simplify the equation and solve for the variable.

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. In other words, a logarithmic equation is the inverse of an exponential equation.

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

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

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 apply the change of base formula
Not using the properties of logarithms correctly
Not isolating the logarithm on one side of the equation
Not checking the solution by plugging it back into the original equation

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

A: The choice of base for a logarithmic equation depends on the specific problem and the context in which it is being used. Some common bases include the natural logarithm (base e), the logarithm with base 10, and the logarithm with base 2.

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

A: Logarithmic equations have many real-world applications, including:

Modeling population growth and decay
Analyzing financial data and predicting stock prices
Understanding the behavior of complex systems and networks
Solving problems in physics, engineering, and computer science

Q: How do I use logarithmic equations in real-world applications?

A: To use logarithmic equations in real-world applications, you can follow these steps:

Identify the problem and the context in which it is being used.
Choose the appropriate base for the logarithmic equation.
Apply the properties of logarithms and the change of base formula as needed.
Solve the equation and interpret the results in the context of the problem.

Q: What are some common types of logarithmic equations?

A: Some common types of logarithmic equations include:

Logarithmic equations with a single logarithm
Logarithmic equations with multiple logarithms
Logarithmic equations with a change of base
Logarithmic equations with a combination of logarithmic and exponential functions

Q: How do I solve logarithmic equations with multiple logarithms?

A: To solve logarithmic equations with multiple logarithms, you can use the following steps:

Apply the product rule and the quotient rule as needed.
Use the change of base formula to rewrite the equation in terms of a common logarithm.
Isolate the logarithm on one side of the equation.
Solve for the variable.

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

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

Forgetting to apply the product rule and the quotient rule
Not using the change of base formula correctly
Not isolating the logarithm on one side of the equation
Not checking the solution by plugging it back into the original equation.