Solving $4 \cdot 5^{x+6} = 625 \cdot 2^{x+4}$ A Step-by-Step Guide

Introduction

In this article, we will explore how to solve the exponential equation $4 \cdot 5^{x+6} = 625 \cdot 2^{x+4}$. Exponential equations like this one often appear in various mathematical contexts, including algebra, calculus, and real-world applications such as growth and decay models. To effectively solve such equations, we need to employ the properties of exponents and logarithms. The main goal is to isolate the variable x. We will go through a step-by-step process, transforming the equation into a more manageable form, and finally, we will use logarithms to find the value of x. We will also discuss the importance of verifying the solution to ensure its correctness. The methodology used here can be applied to similar exponential equations, making this a valuable skill for anyone studying mathematics or related fields. Furthermore, understanding how to manipulate exponential equations enhances problem-solving abilities and provides a solid foundation for more advanced mathematical concepts. So, let's dive into the solution and uncover the value of x.

Step-by-Step Solution

To solve the exponential equation $4 \cdot 5^{x+6} = 625 \cdot 2^{x+4}$, we will follow a structured approach that involves simplifying the equation, using the properties of exponents, and applying logarithms to isolate the variable x. This process not only helps in finding the solution but also enhances understanding of exponential functions and their manipulations. The steps outlined below will provide a clear pathway to the final answer, ensuring that each transformation is logically sound and mathematically accurate. Let's start by rewriting the equation to make it easier to work with.

1. Rewrite the equation: Begin by expressing all terms as powers of their prime factors. We have $4 = 2^2$ and $625 = 5^4$. Substitute these into the original equation:

   $2^2 \cdot 5^{x+6} = 5^4 \cdot 2^{x+4}$

2. Rearrange the terms: Group the powers of 2 and 5 on each side of the equation. Divide both sides by $2^{x+4}$ and $5^{x+6}$ to get:

   $\frac{5^{x+6}}{5^4} = \frac{2^{x+4}}{2^2}$

3. Apply exponent rules: Use the quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$. Apply this rule to both sides:

   $5^{(x+6)-4} = 2^{(x+4)-2}$

   Simplify the exponents:

   $5^{x+2} = 2^{x+2}$

4. Isolate the variable: Now, we have an equation where the bases are different, but the exponents are the same. To proceed, divide both sides by $2^{x+2}$:

   $\frac{5^{x+2}}{2^{x+2}} = 1$

5. Combine exponents: Use the rule $\frac{a^n}{b^n} = (\frac{a}{b})^n$ to combine the terms on the left side:

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

6. Solve for x: Recognize that any non-zero number raised to the power of 0 equals 1. Therefore:

   $x + 2 = 0$

   Subtract 2 from both sides:

   $x = -2$

7. Verification: Substitute the value of $x$ back into the original equation to verify the solution:

   $4 \cdot 5^{-2+6} = 625 \cdot 2^{-2+4}$

   $4 \cdot 5^4 = 625 \cdot 2^2$

   $4 \cdot 625 = 625 \cdot 4$

   $2500 = 2500$

   Since the equation holds true, our solution is correct.

8. Final Answer: The value of $x$ is -2.

Alternative Method Using Logarithms

Another effective method to solve the equation $4 \cdot 5^{x+6} = 625 \cdot 2^{x+4}$ involves the use of logarithms. Logarithms are particularly useful when dealing with exponential equations where it is difficult to express both sides with a common base. This approach allows us to bring the variable x down from the exponent, making it easier to isolate and solve for its value. By applying logarithmic properties, we can transform the equation into a linear form, which is much simpler to handle. This method also highlights the versatility of logarithms in solving various types of exponential equations. Let's walk through the steps of using logarithms to find the solution.

1. Rewrite the equation: As before, express all terms as powers of their prime factors. We have $4 = 2^2$ and $625 = 5^4$. Substitute these into the original equation:

   $2^2 \cdot 5^{x+6} = 5^4 \cdot 2^{x+4}$

2. Apply logarithms: Take the logarithm of both sides of the equation. We can use any base for the logarithm, but the common logarithm (base 10) or the natural logarithm (base e) are frequently used. Here, we will use the natural logarithm (ln):

   $\ln(2^2 \cdot 5^{x+6}) = \ln(5^4 \cdot 2^{x+4})$

3. Use logarithm properties: Apply the logarithm product rule, which states that $\ln(ab) = \ln(a) + \ln(b)$, and the power rule, which states that $\ln(a^b) = b \cdot \ln(a)$:

   $\ln(2^2) + \ln(5^{x+6}) = \ln(5^4) + \ln(2^{x+4})$

   $2\ln(2) + (x+6)\ln(5) = 4\ln(5) + (x+4)\ln(2)$

4. Expand and rearrange: Distribute and rearrange the terms to isolate x:

   $2\ln(2) + x\ln(5) + 6\ln(5) = 4\ln(5) + x\ln(2) + 4\ln(2)$

5. Collect like terms: Group the terms containing x on one side and the constants on the other side:

   $x\ln(5) - x\ln(2) = 4\ln(2) - 2\ln(2) + 4\ln(5) - 6\ln(5)$

   $x(\ln(5) - \ln(2)) = 2\ln(2) - 2\ln(5)$

6. Solve for x: Divide both sides by $(\ln(5) - \ln(2))$ to solve for x:

   $x = \frac{2\ln(2) - 2\ln(5)}{\ln(5) - \ln(2)}$

7. Simplify: Factor out a 2 from the numerator:

   $x = 2 \cdot \frac{\ln(2) - \ln(5)}{\ln(5) - \ln(2)}$

   Notice that the numerator is the negative of the denominator, so:

   $x = 2 \cdot (-1)$

   $x = -2$

8. Verification: As shown in the previous method, substitute the value of $x$ back into the original equation to verify the solution. The verification step confirms that our solution is correct.

9. Final Answer: The value of $x$ is -2.

Conclusion

In conclusion, we have successfully solved the exponential equation $4 \cdot 5^{x+6} = 625 \cdot 2^{x+4}$ using two different methods: simplifying and equating exponents, and applying logarithms. Both approaches led us to the same solution, $x = -2$. The first method involved rewriting the equation with prime factors, applying exponent rules, and simplifying to isolate x. This method is straightforward and efficient when the terms can be easily expressed as powers of the same base. The second method utilized logarithms, which are particularly useful when dealing with exponential equations that are not easily simplified using exponent rules alone. By taking the logarithm of both sides, we transformed the equation into a linear form, making it easier to solve for x. This approach demonstrates the power and versatility of logarithms in solving exponential equations. The verification step in both methods confirmed the correctness of our solution, reinforcing the importance of checking our work in mathematical problem-solving. Understanding these methods not only enhances our ability to solve exponential equations but also deepens our grasp of the fundamental principles of exponents and logarithms. The skills acquired through this process are invaluable in various mathematical and scientific contexts, making the effort to master these techniques well worth it.