Select The Correct Answer. Solve The Exponential Equation For X.

Select the Correct Answer: Solving Exponential Equations for x

Solving exponential equations is a crucial skill in algebra and numerous applications across science and engineering. These equations involve variables in the exponent, requiring specific techniques to isolate and solve for the unknown. This comprehensive guide will delve into various methods for solving exponential equations, providing numerous examples and clarifying common pitfalls. We'll explore techniques applicable to different types of exponential equations, empowering you to select the correct answer with confidence.

Understanding Exponential Equations

Before diving into solution methods, let's define what an exponential equation is. An exponential equation is an equation where the variable appears in the exponent. The general form is:

a^x = b

where 'a' and 'b' are constants, and 'x' is the variable we aim to solve for. The base, 'a', is typically a positive number other than 1.

Methods for Solving Exponential Equations

Several methods can be used to solve exponential equations, depending on the complexity of the equation. Here are some of the most common techniques:

1. Using the Property of Equality for Exponential Functions

If the bases are equal, then the exponents must also be equal. This is the simplest method and applies when both sides of the equation have the same base raised to different exponents. For example:

2^x = 2⁵

Since the bases (2) are the same, we can equate the exponents:

x = 5

2. Rewriting with a Common Base

This technique is useful when the bases are different but can be expressed as powers of the same base. For instance:

4^x = 8

We can rewrite 4 as 2² and 8 as 2³. Substituting these into the equation gives:

(2²)^x = 2³

Using the power of a power rule ((a^m)ⁿ = a^mn), we get:

2^2x = 2³

Now, the bases are the same, so we equate the exponents:

2x = 3

x = 3/2

3. Using Logarithms

When the bases cannot be easily made the same, logarithms are the most powerful tool. Logarithms are the inverse function of exponentials. The key property we use is:

log_a(a^x) = x

Let's illustrate this with an example:

3^x = 10

To solve for x, we take the logarithm of both sides. We can use any base for the logarithm, but base 10 (common logarithm) or base e (natural logarithm) are common choices. Using the natural logarithm (ln):

ln(3^x) = ln(10)

Using the logarithm power rule (log_a(b^c) = c * log_a(b)):

x * ln(3) = ln(10)

Solving for x:

x = ln(10) / ln(3)

This gives an approximate value for x using a calculator.

4. Solving Exponential Equations with Multiple Terms

Equations may involve multiple terms with exponential expressions. These often require algebraic manipulation before applying logarithmic or base-matching techniques. For example:

2^x + 2^x+1 = 12

We can factor out 2^x:

2^x(1 + 2) = 12

2^x(3) = 12

2^x = 4

2^x = 2²

x = 2

5. Solving Exponential Equations with More Complex Exponents

Some exponential equations have more intricate exponents, involving multiple variables or expressions. Consider:

e^2x+1 = 5

Taking the natural logarithm of both sides:

ln(e^2x+1) = ln(5)

2x + 1 = ln(5)

2x = ln(5) - 1

x = (ln(5) - 1) / 2

Common Mistakes to Avoid

Several common errors can hinder your ability to solve exponential equations correctly. These include:

• Incorrectly applying logarithm rules: Remember the power rule and other logarithmic properties. Incorrect application leads to inaccurate solutions.
• Forgetting to take the logarithm of both sides: When employing logarithms, remember to apply the logarithm to both sides of the equation to maintain balance.
• Mixing up bases: Make sure to use consistent bases throughout your calculations, especially when rewriting expressions with a common base.
• Incorrectly simplifying expressions: Careful simplification is crucial, especially when dealing with multiple terms or complex exponents.

Practice Problems

Let's test your understanding with a series of practice problems:

1. 2^x = 16
2. 3^x = 1/27
3. 5^x = 125
4. 4^x+1 = 64
5. e^x = 7
6. 10^x = 1000
7. 2^x - 2^x-1 = 6
8. 3^2x + 3^x - 2 = 0 (Hint: Treat this as a quadratic equation where y = 3^x)
9. e^-x = 0.2
10. (1/2)^x = 8

Solutions:

1. x = 4
2. x = -3
3. x = 3
4. x = 2
5. x = ln(7)
6. x = 3
7. x = 3
8. x = 0
9. x = ln(5)
10. x = -3

Conclusion

Solving exponential equations is a fundamental skill with wide-ranging applications. By mastering the techniques outlined in this guide and diligently practicing, you'll develop the confidence and proficiency to tackle a variety of exponential equations, accurately selecting the correct answer each time. Remember to pay close attention to detail and carefully review the common pitfalls to avoid errors in your calculations. Consistent practice is key to success in mastering this important algebraic concept.