Solve For { X $} : : : { 10^x - 8 = 92 \}

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. In this article, we will focus on solving the equation $10^x - 8 = 92$ to find the value of $x$. We will break down the solution into manageable steps, using a combination of algebraic manipulations and properties of exponents.

Understanding the Equation

The given equation is $10^x - 8 = 92$. Our goal is to isolate the variable $x$ and find its value. To do this, we need to manipulate the equation using algebraic operations and properties of exponents.

Step 1: Add 8 to Both Sides

The first step is to add 8 to both sides of the equation to get rid of the negative term. This gives us:

$$10^x = 92 + 8$$

$$10^x = 100$$

Step 2: Use the Definition of Exponentiation

The next step is to use the definition of exponentiation to rewrite the equation. We know that $10^x$ means $10$ raised to the power of $x$. Therefore, we can rewrite the equation as:

$$10^x = 10^2$$

Step 3: Equate the Exponents

Since the bases are the same, we can equate the exponents:

$$x = 2$$

Conclusion

In this article, we solved the equation $10^x - 8 = 92$ to find the value of $x$. We broke down the solution into manageable steps, using a combination of algebraic manipulations and properties of exponents. The final answer is $x = 2$.

Properties of Exponents

Exponents have several properties that are essential in solving exponential equations. Some of the key properties include:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$
• Power of a Power: $(a^m)^n = a^{m \cdot n}$
• Power of a Product: $(ab)^m = a^m \cdot b^m$
• Zero Exponent: $a^0 = 1$

Algebraic Manipulations

Algebraic manipulations are essential in solving exponential equations. Some of the key manipulations include:

• Addition and Subtraction: $a + b = c$ and $a - b = c$
• Multiplication and Division: $a \cdot b = c$ and $a \div b = c$
• Distributive Property: $a(b + c) = ab + ac$

Real-World Applications

Exponential equations have numerous real-world applications in fields such as:

• Finance: Exponential growth and decay are used to model population growth, compound interest, and depreciation.
• Biology: Exponential growth and decay are used to model population growth, disease spread, and chemical reactions.
• Physics: Exponential growth and decay are used to model radioactive decay, sound waves, and electromagnetic waves.

Conclusion

Introduction

In our previous article, we solved the equation $10^x - 8 = 92$ to find the value of $x$. In this article, we will provide a Q&A guide to help you understand the concepts and techniques used in solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, $10^x$ is an exponential expression, where $10$ is the base and $x$ is the exponent.

Q: What are the properties of exponents?

A: The properties of exponents are:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$
• Power of a Power: $(a^m)^n = a^{m \cdot n}$
• Power of a Product: $(ab)^m = a^m \cdot b^m$
• Zero Exponent: $a^0 = 1$

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable (in this case, $x$) by using algebraic manipulations and properties of exponents. Here are the steps:

1. Add or subtract the same value to both sides: This will help you get rid of any constants on the same side as the exponential expression.
2. Use the definition of exponentiation: This will help you rewrite the equation in a more manageable form.
3. Equate the exponents: Since the bases are the same, you can equate the exponents to find the value of the variable.

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

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

• Not using the correct properties of exponents: Make sure you use the correct properties of exponents to simplify the equation.
• Not isolating the variable: Make sure you isolate the variable (in this case, $x$) by using algebraic manipulations and properties of exponents.
• Not checking your work: Make sure you check your work by plugging the solution back into the original equation.

Q: How do I apply exponential equations in real-world situations?

A: Exponential equations have numerous real-world applications in fields such as:

• Finance: Exponential growth and decay are used to model population growth, compound interest, and depreciation.
• Biology: Exponential growth and decay are used to model population growth, disease spread, and chemical reactions.
• Physics: Exponential growth and decay are used to model radioactive decay, sound waves, and electromagnetic waves.

Q: What are some examples of exponential equations in real-world situations?

A: Some examples of exponential equations in real-world situations include:

• Population growth: The population of a city grows exponentially, with a growth rate of 2% per year.
• Compound interest: A savings account earns interest at a rate of 5% per year, compounded annually.
• Radioactive decay: A radioactive substance decays exponentially, with a half-life of 10 years.

Conclusion

In conclusion, solving exponential equations requires a deep understanding of algebraic manipulations and properties of exponents. By following the steps outlined in this article and avoiding common mistakes, you can solve exponential equations and apply them in real-world situations.