What Values Of B Satisfy 4 3b 2 2 64

What Values of b Satisfy 4^3b+2 = 64? A Comprehensive Exploration

This article delves into the mathematical problem of finding the values of 'b' that satisfy the equation 4^3b+2 = 64. We'll explore various methods to solve this exponential equation, providing a detailed step-by-step approach suitable for students of all levels. Furthermore, we'll discuss the underlying mathematical concepts and provide supplementary examples to solidify your understanding.

Understanding Exponential Equations

Before we jump into solving the equation, let's refresh our understanding of exponential equations. An exponential equation is an equation where the variable appears in the exponent. These equations often involve manipulating exponents using properties of exponents and logarithms. Key properties we'll use include:

• Base Conversion: If a^x = b^y, and the bases 'a' and 'b' can be expressed with a common base, this can simplify the equation considerably.
• Logarithms: Logarithms are the inverse operations of exponentiation. Applying logarithms allows us to "bring down" the exponents, simplifying the equations. The change of base formula, log_ab = log_cb/log_ca, is a useful tool for handling logarithms of different bases.

Solving 4^3b+2 = 64: Method 1 – Base Conversion

The most straightforward method for solving 4^3b+2 = 64 involves converting both the base and the right-hand side to a common base. In this case, both 4 and 64 are powers of 2:

• 4 = 2²
• 64 = 2⁶

Substituting these into the original equation, we get:

(2²)^3b+2 = 2⁶

Using the power of a power rule ((a^m)ⁿ = a^mn), we simplify the left-hand side:

2^2(3b+2) = 2⁶

Since the bases are now equal, we can equate the exponents:

2(3b + 2) = 6

Now we have a linear equation which we can solve for 'b':

6b + 4 = 6 6b = 2 b = 2/6 b = 1/3

Therefore, using the base conversion method, the value of 'b' that satisfies the equation is b = 1/3.

Solving 4^3b+2 = 64: Method 2 – Using Logarithms

An alternative method involves using logarithms. We can take the logarithm of both sides of the equation. While any base logarithm will work, we'll use base 10 for simplicity:

log(4^3b+2) = log(64)

Using the logarithm power rule (log(a^x) = x log(a)), we get:

(3b + 2)log(4) = log(64)

Now, we can isolate 'b':

3b + 2 = log(64) / log(4)

Using logarithm properties, or a calculator, we find:

log(64) / log(4) = 3 (because 4³ = 64)

Substituting this back into the equation:

3b + 2 = 3 3b = 1 b = 1/3

This confirms our previous result: the value of 'b' that satisfies the equation is b = 1/3.

Verifying the Solution

It's always a good practice to verify our solution. Let's substitute b = 1/3 back into the original equation:

4^3(1/3)+2 = 4¹⁺² = 4³ = 64

The equation holds true, confirming that b = 1/3 is indeed the correct solution.

Further Exploration: Similar Problems and Extensions

The techniques used to solve 4^3b+2 = 64 can be applied to a broader range of exponential equations. Let's consider some variations:

Example 1: Solve 9^2x-1 = 27

Both 9 and 27 are powers of 3 (9 = 3² and 27 = 3³). Following the base conversion method:

(3²)^2x-1 = 3³ 3^4x-2 = 3³ 4x - 2 = 3 4x = 5 x = 5/4

Example 2: Solve 5^x+1 = 125

Here, 125 = 5³. Therefore:

5^x+1 = 5³ x + 1 = 3 x = 2

Example 3: Solve 2^x = 15 (using logarithms)

In this case, base conversion isn't directly possible. We use logarithms:

log(2^x) = log(15) x log(2) = log(15) x = log(15) / log(2) ≈ 3.907

These examples demonstrate the adaptability of these methods to a variety of exponential equations. Remember to always check your solutions by substituting them back into the original equation.

Advanced Concepts and Considerations

While the problems discussed above are relatively straightforward, more complex exponential equations may require more advanced techniques, such as:

• Multiple solutions: Some equations might have multiple solutions. These may require factoring or using the quadratic formula if the equation transforms into a quadratic form.
• No real solutions: Not all exponential equations have real number solutions. For instance, an equation like 2^x = -1 has no real solutions since 2 raised to any real power is always positive.
• Complex solutions: Equations may have complex solutions involving imaginary numbers. These often require the use of complex logarithms.
• Graphical solutions: Visualizing the equation using graphing techniques can help understand the behavior of the equation and identify potential solutions, especially for more complex scenarios.

Conclusion

Solving exponential equations like 4^3b+2 = 64 involves a systematic approach, typically utilizing either base conversion or logarithms. The choice of method depends on the specific equation. Understanding the fundamental properties of exponents and logarithms is essential for effectively solving these equations. Always verify your solution and consider the possibility of multiple or complex solutions for more advanced problems. Practice is key to mastering these techniques and expanding your problem-solving abilities in mathematics. The examples and explanations provided here serve as a solid foundation for tackling a wider range of exponential equations. Remember to utilize these techniques strategically based on the specifics of each problem.