Solve The Equation: $ \log _5 X + 2 \log _x 5 = 3 $

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Introduction

In this article, we will delve into the world of logarithms and explore a complex equation involving logarithmic functions. The equation $\log_5 x + 2 \log_x 5 = 3$ may seem daunting at first, but with the right approach and techniques, we can simplify and solve it. We will break down the solution step by step, using various properties and identities of logarithms to arrive at the final answer.

Understanding the Equation

The given equation involves logarithms with different bases, namely $\log_5 x$ and $\log_x 5$. To simplify the equation, we need to apply the properties of logarithms, specifically the change of base formula and the power rule.

Change of Base Formula

The change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $a$, $b$, and $c$ are positive real numbers and $c \neq 1$. We can use this formula to rewrite the equation in terms of a common base.

Power Rule

The power rule states that $\log_b (x^n) = n \log_b x$, where $x$ is a positive real number and $n$ is a real number. We can use this rule to simplify the equation by combining the logarithmic terms.

Simplifying the Equation

Using the change of base formula, we can rewrite the equation as follows:

$$\frac{\log x}{\log 5} + 2 \frac{\log 5}{\log x} = 3$$

Now, we can apply the power rule to simplify the equation further:

$$\frac{\log x}{\log 5} + \frac{2 \log 5}{\log x} = 3$$

Combining Logarithmic Terms

We can combine the logarithmic terms by finding a common denominator:

$$\frac{\log x}{\log 5} + \frac{2 \log 5}{\log x} = \frac{(\log x)^2 + 2 (\log 5)^2}{(\log 5) (\log x)}$$

Simplifying the Expression

We can simplify the expression by canceling out the common factors:

$$\frac{(\log x)^2 + 2 (\log 5)^2}{(\log 5) (\log x)} = \frac{(\log x)^2 + 2 (\log 5)^2}{\log 5 \log x}$$

Solving for x

We can now solve for $x$ by equating the numerator and denominator:

$$(\log x)^2 + 2 (\log 5)^2 = 3 \log 5 \log x$$

Expanding and Simplifying

We can expand and simplify the equation by distributing the terms:

$$(\log x)^2 + 2 (\log 5)^2 = 3 \log 5 \log x$$

$$(\log x)^2 - 3 \log 5 \log x + 2 (\log 5)^2 = 0$$

Factoring the Quadratic

We can factor the quadratic equation as follows:

$$(\log x - 2 \log 5)(\log x - \log 5) = 0$$

Solving for x

We can now solve for $x$ by equating each factor to zero:

$$\log x - 2 \log 5 = 0$$

$$\log x - \log 5 = 0$$

Solving the First Equation

We can solve the first equation by adding $2 \log 5$ to both sides:

$$\log x = 2 \log 5$$

Exponentiating Both Sides

We can exponentiate both sides to get rid of the logarithm:

$$x = 5^2$$

Solving the Second Equation

We can solve the second equation by adding $\log 5$ to both sides:

$$\log x = \log 5$$

Exponentiating Both Sides

We can exponentiate both sides to get rid of the logarithm:

$$x = 5$$

Conclusion

In this article, we have solved the equation $\log_5 x + 2 \log_x 5 = 3$ using various properties and identities of logarithms. We have broken down the solution step by step, simplifying the equation and arriving at the final answer. The two possible solutions are $x = 25$ and $x = 5$.

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Introduction

In our previous article, we solved the equation $\log_5 x + 2 \log_x 5 = 3$ using various properties and identities of logarithms. In this article, we will answer some frequently asked questions related to the solution of this equation.

Q: What is the change of base formula and how is it used in solving the equation?

A: The change of base formula is a property of logarithms that allows us to rewrite a logarithmic expression in terms of a common base. In the equation $\log_5 x + 2 \log_x 5 = 3$, we used the change of base formula to rewrite the logarithmic expressions in terms of the natural logarithm.

Q: What is the power rule and how is it used in solving the equation?

A: The power rule is a property of logarithms that states that $\log_b (x^n) = n \log_b x$. In the equation $\log_5 x + 2 \log_x 5 = 3$, we used the power rule to simplify the logarithmic expressions and combine them into a single expression.

Q: How do we simplify the equation $\frac{\log x}{\log 5} + \frac{2 \log 5}{\log x} = 3$?

A: To simplify the equation, we can combine the logarithmic terms by finding a common denominator. This allows us to rewrite the equation as $\frac{(\log x)^2 + 2 (\log 5)^2}{(\log 5) (\log x)} = 3$.

Q: How do we solve for x in the equation $(\log x)^2 + 2 (\log 5)^2 = 3 \log 5 \log x$?

A: To solve for x, we can expand and simplify the equation by distributing the terms. This allows us to rewrite the equation as $(\log x)^2 - 3 \log 5 \log x + 2 (\log 5)^2 = 0$. We can then factor the quadratic equation and solve for x.

Q: What are the two possible solutions to the equation $\log_5 x + 2 \log_x 5 = 3$?

A: The two possible solutions to the equation are $x = 25$ and $x = 5$.

Q: How do we verify the solutions to the equation?

A: To verify the solutions, we can substitute each solution back into the original equation and check if it is true. If the equation holds true for a particular solution, then that solution is valid.

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 properties and identities of logarithms
• Not simplifying the equation correctly
• Not checking the solutions for validity
• Not using the correct notation and terminology

Conclusion

In this article, we have answered some frequently asked questions related to the solution of the equation $\log_5 x + 2 \log_x 5 = 3$. We have provided explanations and examples to help clarify the concepts and techniques used in solving the equation. By following the steps and tips outlined in this article, you can improve your understanding and skills in solving logarithmic equations.

Additional Resources

• For more information on logarithmic equations, see our previous article on solving the equation $\log_5 x + 2 \log_x 5 = 3$.
• For practice problems and exercises, see our logarithmic equation worksheet.
• For more resources and tutorials, see our logarithmic equation tutorial page.

Final Thoughts

Solving logarithmic equations can be challenging, but with practice and patience, you can develop the skills and confidence to tackle even the most complex equations. Remember to use the correct properties and identities of logarithms, simplify the equation correctly, and check the solutions for validity. With these tips and techniques, you can become proficient in solving logarithmic equations and apply them to real-world problems.