Solve For { X$}$ In The Equation:${10^{5x} + 2 = 51}$

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Introduction

In this article, we will delve into solving for $x$ in the equation $10^{5x} + 2 = 51$. This equation involves an exponential term and a constant term, making it a bit more complex than a standard linear equation. We will use algebraic manipulations and properties of exponents to isolate the variable $x$ and find its value.

Understanding the Equation

The given equation is $10^{5x} + 2 = 51$. The first step is to understand the structure of the equation and identify the key components. We have an exponential term $10^{5x}$ and a constant term $2$ on the left-hand side, and a constant term $51$ on the right-hand side.

Isolating the Exponential Term

To solve for $x$, we need to isolate the exponential term $10^{5x}$. We can do this by subtracting $2$ from both sides of the equation:

$$10^{5x} + 2 - 2 = 51 - 2$$

This simplifies to:

$$10^{5x} = 49$$

Using Properties of Exponents

Now that we have isolated the exponential term, we can use properties of exponents to simplify the equation further. We know that $49$ can be expressed as $7^2$. Therefore, we can rewrite the equation as:

$$10^{5x} = 7^2$$

Applying the Property of Exponents

We can use the property of exponents that states $a^b = c^d$ implies $b \log a = d \log c$. Applying this property to our equation, we get:

$$5x \log 10 = 2 \log 7$$

Simplifying the Equation

We can simplify the equation further by using the fact that $\log 10 = 1$. Therefore, we can rewrite the equation as:

$$5x = 2 \log 7$$

Solving for $x$

Now that we have simplified the equation, we can solve for $x$. We can do this by dividing both sides of the equation by $5$:

$$x = \frac{2 \log 7}{5}$$

Evaluating the Expression

To find the value of $x$, we need to evaluate the expression $\frac{2 \log 7}{5}$. We can use a calculator to find the value of $\log 7$ and then plug it into the expression.

Conclusion

In this article, we solved for $x$ in the equation $10^{5x} + 2 = 51$. We used algebraic manipulations and properties of exponents to isolate the variable $x$ and find its value. The final answer is $x = \frac{2 \log 7}{5}$.

Final Answer

The final answer is $\boxed{\frac{2 \log 7}{5}}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. $10^{5x} + 2 = 51$
2. $10^{5x} = 49$
3. $10^{5x} = 7^2$
4. $5x \log 10 = 2 \log 7$
5. $5x = 2 \log 7$
6. $x = \frac{2 \log 7}{5}$

Frequently Asked Questions

• Q: What is the value of $x$ in the equation $10^{5x} + 2 = 51$? A: The value of $x$ is $\frac{2 \log 7}{5}$.
• Q: How do we solve for $x$ in the equation $10^{5x} + 2 = 51$? A: We use algebraic manipulations and properties of exponents to isolate the variable $x$ and find its value.
• Q: What is the final answer to the problem? A: The final answer is $\boxed{\frac{2 \log 7}{5}}$.

Related Problems

• Solve for $x$ in the equation $10^{3x} + 5 = 67$
• Solve for $x$ in the equation $10^{2x} + 3 = 53$
• Solve for $x$ in the equation $10^{4x} + 2 = 59$

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  

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Introduction

In our previous article, we solved for $x$ in the equation $10^{5x} + 2 = 51$. We used algebraic manipulations and properties of exponents to isolate the variable $x$ and find its value. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the value of $x$ in the equation $10^{5x} + 2 = 51$?

A: The value of $x$ is $\frac{2 \log 7}{5}$.

Q: How do we solve for $x$ in the equation $10^{5x} + 2 = 51$?

A: We use algebraic manipulations and properties of exponents to isolate the variable $x$ and find its value.

Q: What is the final answer to the problem?

A: The final answer is $\boxed{\frac{2 \log 7}{5}}$.

Q: Can you explain the steps to solve for $x$ in the equation $10^{5x} + 2 = 51$?

A: Here are the steps to solve for $x$ in the equation $10^{5x} + 2 = 51$:

1. $10^{5x} + 2 = 51$
2. $10^{5x} = 49$
3. $10^{5x} = 7^2$
4. $5x \log 10 = 2 \log 7$
5. $5x = 2 \log 7$
6. $x = \frac{2 \log 7}{5}$

Q: What is the relationship between the equation $10^{5x} + 2 = 51$ and the equation $10^{3x} + 5 = 67$?

A: The two equations are similar, but with different coefficients and constants. To solve for $x$ in the equation $10^{3x} + 5 = 67$, we can use the same steps as before, but with the new coefficients and constants.

Q: Can you provide more examples of equations that can be solved using the same steps as the equation $10^{5x} + 2 = 51$?

A: Yes, here are some examples of equations that can be solved using the same steps as the equation $10^{5x} + 2 = 51$:

• $10^{2x} + 3 = 53$
• $10^{4x} + 2 = 59$
• $10^{6x} + 5 = 71$

Q: How do we evaluate the expression $\frac{2 \log 7}{5}$?

A: To evaluate the expression $\frac{2 \log 7}{5}$, we can use a calculator to find the value of $\log 7$ and then plug it into the expression.

Conclusion

In this article, we answered some frequently asked questions related to the problem of solving for $x$ in the equation $10^{5x} + 2 = 51$. We provided step-by-step solutions to the problem and answered questions about the relationship between the equation and other similar equations.

Final Answer

The final answer is $\boxed{\frac{2 \log 7}{5}}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. $10^{5x} + 2 = 51$
2. $10^{5x} = 49$
3. $10^{5x} = 7^2$
4. $5x \log 10 = 2 \log 7$
5. $5x = 2 \log 7$
6. $x = \frac{2 \log 7}{5}$

Frequently Asked Questions

• Q: What is the value of $x$ in the equation $10^{5x} + 2 = 51$? A: The value of $x$ is $\frac{2 \log 7}{5}$.
• Q: How do we solve for $x$ in the equation $10^{5x} + 2 = 51$? A: We use algebraic manipulations and properties of exponents to isolate the variable $x$ and find its value.
• Q: What is the final answer to the problem? A: The final answer is $\boxed{\frac{2 \log 7}{5}}$.

Related Problems

• Solve for $x$ in the equation $10^{3x} + 5 = 67$
• Solve for $x$ in the equation $10^{2x} + 3 = 53$
• Solve for $x$ in the equation $10^{4x} + 2 = 59$

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton