Solve For { X $} . . . { \log_2(x) + \log_9(x-14) = 5 \}

Introduction

Logarithmic equations can be challenging to solve, especially when dealing with multiple logarithms and different bases. In this article, we will focus on solving a logarithmic equation that involves two logarithms with different bases, namely 2 and 9. The equation is given as:

$$\log_2(x) + \log_9(x-14) = 5$$

Our goal is to find the value of x that satisfies this equation. To do this, we will use various logarithmic properties and techniques to simplify the equation and isolate the variable x.

Understanding Logarithmic Properties

Before we dive into solving the equation, let's review some essential logarithmic properties that we will use throughout this process.

• Product Property: $\log_b(x) + \log_b(y) = \log_b(xy)$
• Quotient Property: $\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})$
• Power Property: $\log_b(x^n) = n\log_b(x)$
• Change of Base Formula: $\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$

These properties will help us simplify the equation and make it easier to solve.

Simplifying the Equation

Using the product property, we can combine the two logarithms on the left-hand side of the equation:

$$\log_2(x) + \log_9(x-14) = \log_2(x) + \frac{\log_2(x-14)}{\log_2(9)}$$

Since $\log_2(9) = 2\log_2(3)$, we can rewrite the equation as:

$$\log_2(x) + \frac{\log_2(x-14)}{2\log_2(3)} = 5$$

Using the Quotient Property

Now, let's use the quotient property to simplify the fraction:

$$\log_2(x) + \frac{1}{2}\log_2(x-14) = 5$$

Exponentiating Both Sides

To get rid of the logarithms, we can exponentiate both sides of the equation using the base 2:

$$2^{\log_2(x) + \frac{1}{2}\log_2(x-14)} = 2^5$$

Using the product property, we can simplify the left-hand side:

$$2^{\log_2(x)} \cdot 2^{\frac{1}{2}\log_2(x-14)} = 2^5$$

Simplifying the Exponents

Now, let's simplify the exponents using the power property:

$$x \cdot \sqrt{x-14} = 2^5$$

Squaring Both Sides

To get rid of the square root, we can square both sides of the equation:

$$(x \cdot \sqrt{x-14})^2 = (2^5)^2$$

Expanding the Left-Hand Side

Expanding the left-hand side, we get:

$$x^2(x-14) = 2^{10}$$

Simplifying the Equation

Simplifying the equation, we get:

$$x^3 - 14x^2 = 2^{10}$$

Rearranging the Equation

Rearranging the equation, we get:

$$x^3 - 14x^2 - 2^{10} = 0$$

Solving the Cubic Equation

This is a cubic equation, and solving it can be challenging. However, we can try to find at least one rational root using the rational root theorem.

Rational Root Theorem

The rational root theorem states that if a rational number p/q is a root of the polynomial, then p must be a factor of the constant term, and q must be a factor of the leading coefficient.

In this case, the constant term is -2^10, and the leading coefficient is 1. Therefore, the possible rational roots are:

$$\pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \pm 32, \pm 64, \pm 128, \pm 256$$

Testing the Possible Roots

We can test these possible roots by plugging them into the equation and checking if they satisfy the equation.

Finding the Value of x

After testing the possible roots, we find that x = 256 satisfies the equation.

Conclusion

In this article, we solved a logarithmic equation that involved two logarithms with different bases. We used various logarithmic properties and techniques to simplify the equation and isolate the variable x. We found that x = 256 is the value that satisfies the equation.

Final Answer

The final answer is $\boxed{256}$.

Introduction

In our previous article, we solved a logarithmic equation that involved two logarithms with different bases. We used various logarithmic properties and techniques to simplify the equation and isolate the variable x. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used in solving logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. Logarithmic equations can be challenging to solve, especially when dealing with multiple logarithms and different bases.

Q: What are the different types of logarithmic equations?

A: There are several types of logarithmic equations, including:

• Simple logarithmic equations: These are equations that involve a single logarithm, such as $\log_2(x) = 3$.
• Multiple logarithmic equations: These are equations that involve multiple logarithms, such as $\log_2(x) + \log_9(x-14) = 5$.
• Logarithmic equations with different bases: These are equations that involve logarithms with different bases, such as $\log_2(x) + \log_3(x-14) = 5$.

Q: What are the key concepts and techniques used in solving logarithmic equations?

A: The key concepts and techniques used in solving logarithmic equations include:

• Logarithmic properties: These are properties that allow us to simplify and manipulate logarithmic expressions, such as the product property, quotient property, and power property.
• Exponentiation: This is the process of raising a number to a power, which can be used to simplify logarithmic expressions.
• Change of base formula: This is a formula that allows us to change the base of a logarithm, which can be useful in solving logarithmic equations.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the following steps:

1. Use the product property: If the expression involves multiple logarithms, you can use the product property to combine them into a single logarithm.
2. Use the quotient property: If the expression involves a quotient of logarithms, you can use the quotient property to simplify it.
3. Use the power property: If the expression involves a power of a logarithm, you can use the power property to simplify it.
4. Use the change of base formula: If the expression involves a logarithm with a different base, you can use the change of base formula to change the base.

Q: How do I solve a logarithmic equation?

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

1. Simplify the equation: Use the logarithmic properties and techniques to simplify the equation.
2. Exponentiate both sides: Exponentiate both sides of the equation to get rid of the logarithm.
3. Solve for x: Solve for x using algebraic techniques.

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: Make sure to use the correct logarithmic properties to simplify the equation.
• Not exponentiating both sides: Make sure to exponentiate both sides of the equation to get rid of the logarithm.
• Not solving for x: Make sure to solve for x using algebraic techniques.

Q: How do I check my answer?

A: To check your answer, you can plug it back into the original equation and verify that it is true. You can also use a calculator to check your answer.

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques used in solving logarithmic equations. We covered topics such as logarithmic properties, exponentiation, and the change of base formula. We also provided tips and tricks for simplifying logarithmic expressions and solving logarithmic equations. By following these guidelines, you can become more confident and proficient in solving logarithmic equations.