What Is The First Step In Proving That 10 Log ⁡ 100 = 100 10^{\log 100} = 100 1 0 L O G 100 = 100 ?A. Determine What Log ⁡ 100 \log 100 Lo G 100 Is Equal To.B. Divide Both Sides Of The Equation By 10.C. Cancel Out The Exponential Base 10 And The Logarithm.D. Solve The Equation $10^x =

$$

Introduction

In mathematics, proving an equation involves a series of logical steps that demonstrate the truth of the equation. When dealing with exponential and logarithmic functions, it's essential to understand the properties and relationships between these functions. In this article, we will explore the first step in proving that $10^{\log 100} = 100$.

Understanding the Equation

The equation $10^{\log 100} = 100$ involves both exponential and logarithmic functions. The exponential function $10^x$ raises 10 to the power of $x$, while the logarithmic function $\log 100$ represents the power to which 10 must be raised to produce 100. To prove this equation, we need to understand the properties of these functions and how they interact.

The First Step: Determine What $\log 100$ is Equal To

The first step in proving the equation $10^{\log 100} = 100$ is to determine what $\log 100$ is equal to. This involves understanding the definition of a logarithm and how it relates to the exponential function. A logarithm is the inverse of an exponential function, and it represents the power to which a base must be raised to produce a given number.

In this case, we need to find the value of $\log 100$. Since $\log 100$ represents the power to which 10 must be raised to produce 100, we can write:

$$\log 100 = x$$

where $x$ is the power to which 10 must be raised to produce 100.

Properties of Logarithms

To find the value of $\log 100$, we need to understand the properties of logarithms. One of the key properties of logarithms is that they are the inverse of exponential functions. This means that if $a^x = b$, then $\log_a b = x$. In this case, we have:

$$10^x = 100$$

Taking the logarithm of both sides with base 10, we get:

$$\log_{10} 10^x = \log_{10} 100$$

Using the property of logarithms that $\log_a a^x = x$, we can simplify the left-hand side to get:

$$x = \log_{10} 100$$

Evaluating $\log_{10} 100$

To evaluate $\log_{10} 100$, we need to understand that the logarithm is asking for the power to which 10 must be raised to produce 100. Since $10^2 = 100$, we can conclude that:

$$\log_{10} 100 = 2$$

Conclusion

In conclusion, the first step in proving that $10^{\log 100} = 100$ is to determine what $\log 100$ is equal to. This involves understanding the properties of logarithms and how they relate to the exponential function. By evaluating $\log_{10} 100$, we can conclude that $\log 100 = 2$. This is the first step in proving the equation, and it lays the foundation for further steps in the proof.

Further Steps in the Proof

In the next section, we will explore further steps in the proof, including dividing both sides of the equation by 10 and canceling out the exponential base 10 and the logarithm.

Dividing Both Sides of the Equation by 10

Once we have determined that $\log 100 = 2$, we can rewrite the original equation as:

$$10^2 = 100$$

Dividing both sides of the equation by 10, we get:

$$\frac{10^2}{10} = \frac{100}{10}$$

Simplifying both sides, we get:

$$10 = 10$$

This shows that the equation is true, and it provides further evidence for the validity of the original equation.

Canceling Out the Exponential Base 10 and the Logarithm

Another way to prove the equation is to cancel out the exponential base 10 and the logarithm. This involves using the property of logarithms that $\log_a a^x = x$. Since $10^2 = 100$, we can write:

$$\log 10^2 = \log 100$$

Using the property of logarithms that $\log_a a^x = x$, we can simplify the left-hand side to get:

$$2 = \log 100$$

This shows that the equation is true, and it provides further evidence for the validity of the original equation.

Discussion

In this article, we have explored the first step in proving that $10^{\log 100} = 100$. This involves determining what $\log 100$ is equal to, which requires understanding the properties of logarithms and how they relate to the exponential function. By evaluating $\log_{10} 100$, we can conclude that $\log 100 = 2$. This is the first step in proving the equation, and it lays the foundation for further steps in the proof.

Final Thoughts

In conclusion, proving an equation involves a series of logical steps that demonstrate the truth of the equation. When dealing with exponential and logarithmic functions, it's essential to understand the properties and relationships between these functions. By following the steps outlined in this article, we can prove that $10^{\log 100} = 100$.

References

• [1] "Logarithms and Exponents" by Math Open Reference
• [2] "Properties of Logarithms" by Khan Academy
• [3] "Exponential and Logarithmic Functions" by Wolfram MathWorld

Additional Resources

• [1] "Logarithms and Exponents" by MIT OpenCourseWare
• [2] "Properties of Logarithms" by Purplemath
• [3] "Exponential and Logarithmic Functions" by Mathway
  

$$

Introduction

In our previous article, we explored the first step in proving that $10^{\log 100} = 100$. We determined that $\log 100 = 2$ and used this result to prove the equation. In this article, we will answer some common questions related to this proof and provide additional insights into the properties of logarithms and exponential functions.

Q&A

Q: Why is it necessary to determine what $\log 100$ is equal to?

A: Determining what $\log 100$ is equal to is necessary because it allows us to rewrite the original equation in a simpler form. By evaluating $\log_{10} 100$, we can conclude that $\log 100 = 2$, which is a key step in proving the equation.

Q: Can we use other properties of logarithms to prove the equation?

A: Yes, we can use other properties of logarithms to prove the equation. For example, we can use the property that $\log_a a^x = x$ to simplify the left-hand side of the equation. This property allows us to cancel out the exponential base 10 and the logarithm, which provides further evidence for the validity of the original equation.

Q: How do logarithms and exponential functions relate to each other?

A: Logarithms and exponential functions are inverse functions. This means that if $a^x = b$, then $\log_a b = x$. In the case of the equation $10^{\log 100} = 100$, we can use this property to rewrite the equation in a simpler form.

Q: What are some common mistakes to avoid when working with logarithms and exponential functions?

A: Some common mistakes to avoid when working with logarithms and exponential functions include:

• Not understanding the properties of logarithms and exponential functions
• Not evaluating the logarithm correctly
• Not using the correct base for the logarithm
• Not simplifying the equation correctly

Q: Can we use technology to help us prove the equation?

A: Yes, we can use technology to help us prove the equation. For example, we can use a calculator to evaluate the logarithm and simplify the equation. We can also use software such as Mathematica or Maple to help us with the proof.

Q: What are some real-world applications of logarithms and exponential functions?

A: Logarithms and exponential functions have many real-world applications, including:

• Finance: Logarithms and exponential functions are used to calculate interest rates and investment returns.
• Science: Logarithms and exponential functions are used to model population growth and decay.
• Engineering: Logarithms and exponential functions are used to design and optimize systems.

Conclusion

In conclusion, proving that $10^{\log 100} = 100$ involves understanding the properties of logarithms and exponential functions. By following the steps outlined in this article, we can prove the equation and gain a deeper understanding of the relationships between logarithms and exponential functions.

Final Thoughts

In conclusion, logarithms and exponential functions are powerful tools that have many real-world applications. By understanding the properties and relationships between these functions, we can solve a wide range of problems and gain a deeper understanding of the world around us.

References

• [1] "Logarithms and Exponents" by Math Open Reference
• [2] "Properties of Logarithms" by Khan Academy
• [3] "Exponential and Logarithmic Functions" by Wolfram MathWorld

Additional Resources

• [1] "Logarithms and Exponents" by MIT OpenCourseWare
• [2] "Properties of Logarithms" by Purplemath
• [3] "Exponential and Logarithmic Functions" by Mathway