Problem 94 - Intermediate Algebra

Question

Look Alikes \(\cdots\) a. \(5^{9 x-1}=125\) b. \(5^{9 x-1}=124\)

Step-by-Step Solution

Answer

(a) \( x = \frac{4}{9} \); (b) No simple solution.

Step 1: Recognize the Base

Notice that both parts of the problem involve an exponentiation with base 5. For part (a), the equation is a simple exponential equation that can be solved directly.

Step 2: Solve Part (a)

For part (a), we can begin by rewriting the number 125 as a power of 5, i.e., \( 125 = 5^3 \). Therefore, the equation \( 5^{9x-1} = 125 \) becomes \( 5^{9x-1} = 5^3 \). Since the bases are the same, we can equate the exponents: \( 9x - 1 = 3 \). Solve for \( x \) by first adding 1 to both sides to get \( 9x = 4 \) and then dividing by 9 to obtain \( x = \frac{4}{9} \).

Step 3: Direct Observation for Part (b)

For part (b), the equation is \( 5^{9x-1} = 124 \). Since 124 is not a power of 5, it's impossible to express it using an integer power of 5. This suggests no simple solution in integers or rational numbers where the exponent expression on both sides can be equal.

Step 4: Conclusion

In real-world applications, such unsolvable in simple terms equations (like part b) might require numerical methods for approximation. But in algebraic terms, since we can't simplify \( 124 \) to a power of 5, there is no clean algebraic solution for part (b).

Key Concepts

Base Recognition

In exponential equations, recognizing the base plays a crucial role. The base is the number that is repeated in multiplication according to the exponent. For example, in the equation \(5^{9x-1}\), the base is 5, meaning 5 is being multiplied by itself. Identifying the base helps you transform other numbers involved in the equation to the same base to simplify the problem.
In the case of our exercise, we clearly see part (a) and part (b) both contain the base 5. Spotting the base allows us to address each side of the equation in terms of power, especially for part (a) where 125 is recognized and rewritten as \(5^3\). This step is a key starting point in solving many exponential equations.

Exponentiation

Exponentiation is the process of raising a number (the base) to a power. This power denotes the number of times the base is multiplied by itself. For example, in \(5^{3}\), 5 is the base raised to the power of 3, resulting in \(5 \times 5 \times 5 = 125\).
Understanding exponentiation is critical when working with equations like \(5^{9x-1}=125\), because it allows you to reframe numbers like 125 into powers of their respective base. This reframing, as demonstrated in part (a) of our exercise, transforms complex-looking equations into simple linear equations by equating the exponents once the bases on both sides are identical.

Solving Equations

Solving exponential equations requires isolating the variable to find its value. In the exercise solution for part (a), once the bases were the same on each side (both were 5), it allowed the exponents to be set equal to each other: \(9x - 1 = 3\). This conversion simplifies the process to finding the value of the exponent term.
This involves simple algebraic manipulation:

• Add 1 to both sides to isolate the term involving \(x\), resulting in \(9x = 4\).
• Divide both sides by 9 to solve for \(x\), giving \(x = \frac{4}{9}\).

Solving equations ultimately comes down to balancing both sides, focusing on getting the variable alone.

Numerical Methods

Sometimes, as with part (b) of the exercise \(5^{9x-1}=124\), it isn't straightforward to solve the equation algebraically by setting exponents equal due to the number on the right side (124) not being an approachable power of the base (5). In such situations, numerical methods become useful.
Numerical methods include techniques like iteration, approximations, or computational algorithms to find root values or solutions. For equations without a straightforward solution, using numerical methods means employing techniques like the Newton-Raphson method, or other such techniques, to approximate values of \(x\). These methods are especially useful in higher-level mathematics, physics, and engineering where exact algebraic solutions are impractical or impossible. Except in simple educational contexts, numerical methods provide the practical means to find usable answers.