Use The Elementary Properties Of Logarithms To Solve The Following Equation. Write Your Answer As A Fraction Reduced To Lowest Terms.$\log _4(5x) = 2$x = \square$

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will focus on using the elementary properties of logarithms to solve a given equation. We will break down the solution into manageable steps, making it easier for readers to understand and apply the concepts.

Understanding the Properties of Logarithms

Before we dive into solving the equation, let's review the elementary properties of logarithms. These properties are essential in simplifying and solving logarithmic equations.

• Product Property: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient Property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power Property: $\log_b(x^y) = y \cdot \log_b(x)$

Solving the Equation

Now that we have reviewed the properties of logarithms, let's move on to solving the given equation.

$\log _4(5x) = 2$

Our goal is to isolate the variable $x$ and express it as a fraction reduced to lowest terms.

Step 1: Apply the Power Property

We can start by applying the power property to the given equation. Since the logarithm is equal to 2, we can rewrite the equation as:

$\log _4(5x) = \log _4(4^2)$

Using the power property, we can rewrite the right-hand side of the equation as:

$\log _4(5x) = 2 \cdot \log _4(4)$

Step 2: Simplify the Equation

Now that we have applied the power property, let's simplify the equation. We know that $\log _4(4) = 1$, so we can substitute this value into the equation:

$\log _4(5x) = 2 \cdot 1$

Simplifying further, we get:

$\log _4(5x) = 2$

Step 3: Eliminate the Logarithm

To eliminate the logarithm, we can apply the definition of logarithm. The logarithm of a number is the exponent to which the base must be raised to produce that number. In this case, we can rewrite the equation as:

$4^2 = 5x$

Step 4: Solve for x

Now that we have eliminated the logarithm, let's solve for $x$. We can start by simplifying the left-hand side of the equation:

$16 = 5x$

To solve for $x$, we can divide both sides of the equation by 5:

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

Conclusion

In this article, we have used the elementary properties of logarithms to solve a given equation. We have broken down the solution into manageable steps, making it easier for readers to understand and apply the concepts. By applying the power property, simplifying the equation, eliminating the logarithm, and solving for $x$, we have arrived at the final solution:

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

This solution is reduced to lowest terms, as required by the problem statement.

Final Answer

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Introduction

In our previous article, we explored the use of elementary properties of logarithms to solve a given equation. We broke down the solution into manageable steps, making it easier for readers to understand and apply the concepts. In this article, we will continue to delve into the world of logarithmic equations, answering some of the most frequently asked questions.

Q&A

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm. It is a mathematical statement that contains a logarithmic expression, which is an expression that involves a logarithm.

Q: What are the common properties of logarithms?

A: The common properties of logarithms are:

• Product Property: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient Property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power Property: $\log_b(x^y) = y \cdot \log_b(x)$

Q: How do I solve a logarithmic equation?

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

1. Apply the power property: If the logarithm is equal to a power, you can rewrite the equation using the power property.
2. Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary variables.
3. Eliminate the logarithm: Use the definition of logarithm to eliminate the logarithm from the equation.
4. Solve for x: Once the logarithm has been eliminated, solve for x by isolating it on one side of the equation.

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

A: A logarithmic equation and an exponential equation are two different types of equations. A logarithmic equation involves a logarithm, while an exponential equation involves an exponent.

Q: Can you give an example of a logarithmic equation?

A: Yes, here is an example of a logarithmic equation:

$\log _4(5x) = 2$

This equation involves a logarithm and can be solved using the properties of logarithms.

Q: How do I reduce a fraction to lowest terms?

A: To reduce a fraction to lowest terms, you can follow these steps:

1. Find the greatest common divisor (GCD): Find the GCD of the numerator and denominator.
2. Divide both numbers by the GCD: Divide both the numerator and denominator by the GCD.
3. Simplify the fraction: Simplify the fraction by combining like terms.

Q: What is the final answer to the equation $\log _4(5x) = 2$?

A: The final answer to the equation $\log _4(5x) = 2$ is $x = \frac{16}{5}$.

Conclusion

In this article, we have answered some of the most frequently asked questions about logarithmic equations. We have covered topics such as the common properties of logarithms, how to solve a logarithmic equation, and the difference between a logarithmic equation and an exponential equation. By understanding these concepts, you will be better equipped to tackle logarithmic equations and solve them with confidence.

Final Answer

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