Solve The Equation For { X $} : : : { 5 \cdot \left( \log_5 X \right)^2 - \log_5(x^2) = 15 \}

Introduction

In this article, we will delve into solving a logarithmic equation involving a quadratic term. The given equation is $5 \cdot (\log_5 x)^2 - \log_5(x^2) = 15$. Our goal is to isolate the variable $x$ and find its value. We will break down the solution into manageable steps, using properties of logarithms and algebraic manipulations.

Step 1: Simplify the Equation

To begin, let's simplify the equation by using the properties of logarithms. We can rewrite $\log_5(x^2)$ as $2\log_5 x$. Substituting this into the original equation, we get:

$$5 \cdot (\log_5 x)^2 - 2\log_5 x = 15$$

Step 2: Rearrange the Equation

Next, let's rearrange the equation to isolate the quadratic term. We can move the $-2\log_5 x$ term to the right-hand side of the equation:

$$5 \cdot (\log_5 x)^2 = 15 + 2\log_5 x$$

Step 3: Factor Out the Common Term

Now, let's factor out the common term $5$ from the left-hand side of the equation:

$$5 \cdot (\log_5 x)^2 = 5(3 + \frac{2}{5}\log_5 x)$$

Step 4: Simplify the Right-Hand Side

We can simplify the right-hand side of the equation by combining the terms:

$$5 \cdot (\log_5 x)^2 = 5(3 + \frac{2}{5}\log_5 x) = 15 + 2\log_5 x$$

Step 5: Divide Both Sides by 5

To eliminate the coefficient of the quadratic term, let's divide both sides of the equation by $5$:

$$(\log_5 x)^2 = 3 + \frac{2}{5}\log_5 x$$

Step 6: Rearrange the Equation

Now, let's rearrange the equation to isolate the quadratic term:

$$(\log_5 x)^2 - \frac{2}{5}\log_5 x - 3 = 0$$

Step 7: Solve the Quadratic Equation

We can solve the quadratic equation using the quadratic formula:

$$\log_5 x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -\frac{2}{5}$, and $c = -3$. Plugging these values into the quadratic formula, we get:

$$\log_5 x = \frac{\frac{2}{5} \pm \sqrt{\left(-\frac{2}{5}\right)^2 - 4(1)(-3)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$\log_5 x = \frac{\frac{2}{5} \pm \sqrt{\frac{4}{25} + 12}}{2}$$

$$\log_5 x = \frac{\frac{2}{5} \pm \sqrt{\frac{4 + 300}{25}}}{2}$$

$$\log_5 x = \frac{\frac{2}{5} \pm \sqrt{\frac{304}{25}}}{2}$$

$$\log_5 x = \frac{\frac{2}{5} \pm \frac{4\sqrt{19}}{5}}{2}$$

Simplifying the expression further, we get:

$$\log_5 x = \frac{1 \pm 2\sqrt{19}}{5}$$

Step 8: Solve for x

Now that we have the value of $\log_5 x$, we can solve for $x$ by exponentiating both sides of the equation:

$$x = 5^{\frac{1 \pm 2\sqrt{19}}{5}}$$

Conclusion

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Introduction

In our previous article, we solved the equation $5 \cdot (\log_5 x)^2 - \log_5(x^2) = 15$ using properties of logarithms and algebraic manipulations. We broke down the solution into manageable steps, isolating the variable $x$ and finding its value. In this article, we will answer some frequently asked questions (FAQs) related to the solution.

Q: What is the final solution to the equation?

A: The final solution to the equation is $x = 5^{\frac{1 \pm 2\sqrt{19}}{5}}$.

Q: How did you simplify the equation?

A: We simplified the equation by using the properties of logarithms. We rewrote $\log_5(x^2)$ as $2\log_5 x$ and substituted this into the original equation.

Q: What is the significance of the quadratic formula in solving the equation?

A: The quadratic formula is used to solve quadratic equations of the form $ax^2 + bx + c = 0$. In our case, we used the quadratic formula to solve the quadratic equation $(\log_5 x)^2 - \frac{2}{5}\log_5 x - 3 = 0$.

Q: Can you explain the concept of logarithms in more detail?

A: Logarithms are the inverse operation of exponentiation. In other words, if $x = a^y$, then $y = \log_a x$. Logarithms have many applications in mathematics, science, and engineering.

Q: What are some common properties of logarithms?

A: Some common properties of logarithms include:

• $\log_a (xy) = \log_a x + \log_a y$
• $\log_a (x/y) = \log_a x - \log_a y$
• $\log_a (x^n) = n \log_a x$

Q: How do you use logarithms to solve equations?

A: Logarithms can be used to solve equations by rewriting them in a form that involves logarithms. For example, if we have the equation $x^2 = 100$, we can rewrite it as $\log x^2 = \log 100$, which simplifies to $2\log x = 2$. We can then solve for $x$ by dividing both sides by 2.

Q: What are some real-world applications of logarithms?

A: Logarithms have many real-world applications, including:

• Finance: Logarithms are used to calculate interest rates and returns on investment.
• Science: Logarithms are used to measure the intensity of earthquakes and the brightness of stars.
• Engineering: Logarithms are used to design and optimize systems, such as electronic circuits and communication networks.

Conclusion

In this article, we answered some frequently asked questions related to solving the equation $5 \cdot (\log_5 x)^2 - \log_5(x^2) = 15$. We provided explanations and examples to help clarify the concepts and properties of logarithms. We hope this article has been helpful in understanding the solution to the equation and the applications of logarithms in real-world scenarios.