Solve For { X $}$ In The Equation:${ \left(5^x\right) 5^{(x-1)} \equiv 10 }$

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Introduction

In this article, we will delve into solving for $x$ in the given equation $(5^x) 5^{(x-1)} \equiv 10$. This equation involves exponential terms and requires a thorough understanding of exponential properties and logarithmic functions to solve. We will break down the solution step by step, providing a clear and concise explanation of each step.

Understanding the Equation

The given equation is $(5^x) 5^{(x-1)} \equiv 10$. To solve for $x$, we need to simplify the left-hand side of the equation using the properties of exponents. The equation can be rewritten as $5^{x + (x - 1)} \equiv 10$.

Simplifying the Exponents

Using the property of exponents that states $a^m \cdot a^n = a^{m + n}$, we can simplify the left-hand side of the equation. This gives us $5^{2x - 1} \equiv 10$.

Isolating the Exponential Term

To isolate the exponential term, we can take the logarithm of both sides of the equation. We will use the natural logarithm (ln) for this purpose. This gives us $\ln(5^{2x - 1}) \equiv \ln(10)$.

Applying the Logarithmic Property

Using the property of logarithms that states $\ln(a^b) = b \cdot \ln(a)$, we can simplify the left-hand side of the equation. This gives us $(2x - 1) \cdot \ln(5) \equiv \ln(10)$.

Solving for $x$

To solve for $x$, we need to isolate $x$ on one side of the equation. We can do this by dividing both sides of the equation by $2 \cdot \ln(5)$. This gives us $x - \frac{1}{2} \equiv \frac{\ln(10)}{2 \cdot \ln(5)}$.

Simplifying the Right-Hand Side

Using the property of logarithms that states $\ln(a) = \ln(b) \cdot \frac{\ln(a)}{\ln(b)}$, we can simplify the right-hand side of the equation. This gives us $x - \frac{1}{2} \equiv \frac{\ln(10)}{2 \cdot \ln(5)} = \frac{\ln(10)}{\ln(5^2)} = \frac{\ln(10)}{2 \cdot \ln(5)}$.

Isolating $x$

To isolate $x$, we can add $\frac{1}{2}$ to both sides of the equation. This gives us $x \equiv \frac{\ln(10)}{2 \cdot \ln(5)} + \frac{1}{2}$.

Simplifying the Right-Hand Side

Using the property of logarithms that states $\ln(a) = \ln(b) \cdot \frac{\ln(a)}{\ln(b)}$, we can simplify the right-hand side of the equation. This gives us $x \equiv \frac{\ln(10)}{2 \cdot \ln(5)} + \frac{1}{2} = \frac{\ln(10) + \ln(5)}{2 \cdot \ln(5)}$.

Final Solution

Using the property of logarithms that states $\ln(a) + \ln(b) = \ln(a \cdot b)$, we can simplify the right-hand side of the equation. This gives us $x \equiv \frac{\ln(10 \cdot 5)}{2 \cdot \ln(5)} = \frac{\ln(50)}{2 \cdot \ln(5)}$.

Conclusion

In this article, we have solved for $x$ in the equation $(5^x) 5^{(x-1)} \equiv 10$. We have used the properties of exponents and logarithmic functions to simplify the equation and isolate $x$. The final solution is $x \equiv \frac{\ln(50)}{2 \cdot \ln(5)}$.

Final Answer

The final answer is $\boxed{\frac{\ln(50)}{2 \cdot \ln(5)}}$.

Discussion

The solution to this equation involves the use of logarithmic functions and the properties of exponents. The equation can be solved using the natural logarithm (ln) and the property of logarithms that states $\ln(a^b) = b \cdot \ln(a)$. The final solution is a simple expression involving the natural logarithm of 50 divided by twice the natural logarithm of 5.

Related Topics

• Exponential functions
• Logarithmic functions
• Properties of exponents
• Properties of logarithms

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Logarithmic Functions" by Math Open Reference
• [3] "Properties of Exponents" by Math Open Reference
• [4] "Properties of Logarithms" by Math Open Reference
  

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Introduction

In our previous article, we solved for $x$ in the equation $(5^x) 5^{(x-1)} \equiv 10$. In this article, we will answer some frequently asked questions related to the solution of this equation.

Q: What is the main concept used to solve this equation?

A: The main concept used to solve this equation is the property of exponents that states $a^m \cdot a^n = a^{m + n}$. This property allows us to simplify the left-hand side of the equation and isolate the exponential term.

Q: Why do we need to take the logarithm of both sides of the equation?

A: We need to take the logarithm of both sides of the equation to isolate the exponential term. The logarithm allows us to bring the exponent down and simplify the equation.

Q: What is the significance of the natural logarithm (ln) in this solution?

A: The natural logarithm (ln) is used in this solution because it is the base of the natural logarithm that is used in the property of logarithms that states $\ln(a^b) = b \cdot \ln(a)$. This property allows us to simplify the left-hand side of the equation and isolate the exponential term.

Q: How do we simplify the right-hand side of the equation?

A: We simplify the right-hand side of the equation by using the property of logarithms that states $\ln(a) = \ln(b) \cdot \frac{\ln(a)}{\ln(b)}$. This property allows us to simplify the right-hand side of the equation and isolate $x$.

Q: What is the final solution to this equation?

A: The final solution to this equation is $x \equiv \frac{\ln(50)}{2 \cdot \ln(5)}$.

Q: Can this equation be solved using other methods?

A: Yes, this equation can be solved using other methods such as using the property of exponents that states $a^m \cdot a^n = a^{m + n}$ and then using the property of logarithms that states $\ln(a^b) = b \cdot \ln(a)$. However, the method used in this article is the most straightforward and efficient way to solve this equation.

Q: What are some real-world applications of this equation?

A: This equation has many real-world applications in fields such as finance, economics, and engineering. For example, it can be used to model population growth, compound interest, and exponential decay.

Q: Can this equation be solved using a calculator?

A: Yes, this equation can be solved using a calculator. Simply plug in the values of the variables and use the calculator to solve for $x$.

Q: What are some common mistakes to avoid when solving this equation?

A: Some common mistakes to avoid when solving this equation include:

• Not using the property of exponents that states $a^m \cdot a^n = a^{m + n}$ to simplify the left-hand side of the equation.
• Not using the property of logarithms that states $\ln(a^b) = b \cdot \ln(a)$ to simplify the left-hand side of the equation.
• Not isolating the exponential term correctly.
• Not using the correct method to solve the equation.

Conclusion

In this article, we have answered some frequently asked questions related to the solution of the equation $(5^x) 5^{(x-1)} \equiv 10$. We have also discussed some common mistakes to avoid when solving this equation and some real-world applications of this equation.

Final Answer

The final answer is $\boxed{\frac{\ln(50)}{2 \cdot \ln(5)}}$.

Discussion

The solution to this equation involves the use of logarithmic functions and the properties of exponents. The equation can be solved using the natural logarithm (ln) and the property of logarithms that states $\ln(a^b) = b \cdot \ln(a)$. The final solution is a simple expression involving the natural logarithm of 50 divided by twice the natural logarithm of 5.

Related Topics

• Exponential functions
• Logarithmic functions
• Properties of exponents
• Properties of logarithms

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Logarithmic Functions" by Math Open Reference
• [3] "Properties of Exponents" by Math Open Reference
• [4] "Properties of Logarithms" by Math Open Reference