Solve For $x$ In The Logarithmic Equation:$10^{\log \left(10^5\right)} = 10^x$$ X = X = X = [/tex]

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific logarithmic equation involving exponential and logarithmic functions. We will break down the solution step by step, making it easy to understand and follow.

The Logarithmic Equation

The given logarithmic equation is:

$$10^{\log \left(10^5\right)} = 10^x$$

Our goal is to solve for $x$.

Understanding the Equation

Before we dive into the solution, let's understand the equation. The left-hand side of the equation involves a logarithmic function, while the right-hand side involves an exponential function. The logarithmic function is $\log \left(10^5\right)$, which can be rewritten as $\log (10^5)$. This means that the logarithmic function is taking the base 10 logarithm of $10^5$.

Simplifying the Logarithmic Function

To simplify the logarithmic function, we can use the property of logarithms that states $\log_b (b^x) = x$. In this case, we have $\log (10^5)$. Using the property, we can rewrite this as:

$$\log (10^5) = 5$$

This simplifies the logarithmic function to a constant value of 5.

Substituting the Simplified Logarithmic Function

Now that we have simplified the logarithmic function, we can substitute it back into the original equation:

$$10^5 = 10^x$$

Solving for $x$

To solve for $x$, we can use the property of exponential functions that states $a^x = a^y \implies x = y$. In this case, we have $10^5 = 10^x$. Using the property, we can rewrite this as:

$$x = 5$$

Therefore, the solution to the logarithmic equation is $x = 5$.

Conclusion

Solving logarithmic equations can be challenging, but with the right approach, they can be tackled with ease. By simplifying the logarithmic function and using the properties of exponential and logarithmic functions, we can solve for $x$ in the given equation. In this article, we have demonstrated a step-by-step approach to solving logarithmic equations, making it easy to understand and follow.

Common Mistakes to Avoid

When solving logarithmic equations, there are several common mistakes to avoid. These include:

• Not simplifying the logarithmic function: Failing to simplify the logarithmic function can make it difficult to solve the equation.
• Not using the properties of exponential and logarithmic functions: Failing to use the properties of exponential and logarithmic functions can make it difficult to solve the equation.
• Not checking the solution: Failing to check the solution can lead to incorrect answers.

Tips and Tricks

When solving logarithmic equations, there are several tips and tricks to keep in mind. These include:

• Simplifying the logarithmic function: Simplifying the logarithmic function can make it easier to solve the equation.
• Using the properties of exponential and logarithmic functions: Using the properties of exponential and logarithmic functions can make it easier to solve the equation.
• Checking the solution: Checking the solution can help ensure that the answer is correct.

Real-World Applications

Logarithmic equations have several real-world applications. These include:

• Finance: Logarithmic equations are used in finance to calculate interest rates and investment returns.
• Science: Logarithmic equations are used in science to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic equations are used in engineering to calculate the stress and strain on a material.

Conclusion

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Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will provide a Q&A guide to help you understand and solve logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function. The logarithmic function is the inverse of the exponential function. In other words, if $a^x = b$, then $\log_a (b) = x$.

Q: How do I simplify a logarithmic function?

A: To simplify a logarithmic function, you can use the property of logarithms that states $\log_b (b^x) = x$. This means that if the base of the logarithm is the same as the base of the exponent, you can simplify the logarithmic function by rewriting it as the exponent.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of exponential and logarithmic functions. You can also use the fact that if $a^x = b$, then $\log_a (b) = x$. By simplifying the logarithmic function and using these properties, you can solve for $x$.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

• Not simplifying the logarithmic function: Failing to simplify the logarithmic function can make it difficult to solve the equation.
• Not using the properties of exponential and logarithmic functions: Failing to use the properties of exponential and logarithmic functions can make it difficult to solve the equation.
• Not checking the solution: Failing to check the solution can lead to incorrect answers.

Q: What are some real-world applications of logarithmic equations?

A: Logarithmic equations have several real-world applications, including:

• Finance: Logarithmic equations are used in finance to calculate interest rates and investment returns.
• Science: Logarithmic equations are used in science to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic equations are used in engineering to calculate the stress and strain on a material.

Q: How do I check my solution to a logarithmic equation?

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

Q: What are some tips and tricks for solving logarithmic equations?

A: Some tips and tricks for solving logarithmic equations include:

• Simplifying the logarithmic function: Simplifying the logarithmic function can make it easier to solve the equation.
• Using the properties of exponential and logarithmic functions: Using the properties of exponential and logarithmic functions can make it easier to solve the equation.
• Checking the solution: Checking the solution can help ensure that the answer is correct.

Q: Can you provide an example of a logarithmic equation?

A: Yes, here is an example of a logarithmic equation:

$$10^{\log \left(10^5\right)} = 10^x$$

To solve this equation, you can simplify the logarithmic function and use the properties of exponential and logarithmic functions to solve for $x$.

Conclusion

In conclusion, logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. By simplifying the logarithmic function and using the properties of exponential and logarithmic functions, you can solve for $x$ in the given equation. We hope this Q&A guide has been helpful in understanding and solving logarithmic equations.