Mastering Exponential Equations: Solving (x^x)^5 = 100 Step by Step

Exponential equations often appear complex at first glance, but with the right approach, they become much easier to solve. One such intriguing equation is:

 (x^x)^5 = 100

This type of problem is commonly seen in advanced algebra and competitive exams, making it an excellent challenge for math enthusiasts. In this blog, we will break it down into simple steps and find the exact solution.

 

Step-by-Step Solution

Step 1: Rewrite the Equation Using Exponents

We start by simplifying the given expression:

 (x^x)^5 = 100

Using the power rule (a^m)^n = a^{m * n}, we rewrite it as:

x^{5x} = 100

 

Step 2: Convert 100 into an Exponential Form

We express 100 as a power of 10 to make solving easier:

x^{5x} = 10^2

Now, we take the natural logarithm (ln) on both sides to simplify the exponent:

ln(x^{5x}) = ln(10^2)

Using the logarithm property ln(a^b) = b ln a, we get:

5x ln x = 2 ln 10

Since ln 10 approximately 2.302, we rewrite it as:

5x ln x = 4.604

 

Step 3: Solve for x Numerically

At this point, solving for x algebraically is difficult, so we use approximation or numerical methods. By testing values:

• For x = 2:  5(2) ln(2) = 5(2)(0.693) = 6.93 {Too high}
• For x = 1.5:  5(1.5) ln (1.5) = 5(1.5)(0.405) = 3.037 {Too low}
• For x =1.7  5(1.7) ln (1.7) ≈ 4.65
• Through numerical approximation, we find that:

X ≈ 1.7

 

Final Answer

The approximate solution  (x^x)^5 = 100 is:

X ≈ 1.7

 Why This Equation is Important

This problem demonstrates key algebraic concepts, including exponent properties, logarithmic manipulation, and numerical approximation. These techniques are widely used in calculus, engineering, and real-world problem-solving.

 Conclusion

Solving exponential equations like (x^x)^5 = 100 requires a mix of algebraic transformations and numerical methods. By practicing such problems, you can develop a stronger mathematical foundation and enhance your problem-solving skills.

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