Home of the goDam Hoover Dam Alabams

"The United States of America will continue to remember that many who toiled here found their final rest while engaged in the building of this dam". PLAY BALL!! 1 megaton of TNT = 4.184 \times 10^{15} \, \text{Joules} pie real. 4.184 To increase the number 3.14 into the range of 10^4 (which is 10,000 and above), calculate the necessary power of 3.14. Use the equation: 3.14^x \geq 10,000 Solve for x : 1. Take the natural logarithm (ln) of both sides to simplify the calculation: \ln(3.14^x) \geq \ln(10,000) 2. Use the logarithm power rule \ln(a^b) = b \ln(a): x \ln(3.14) \geq \ln(10,000) 3. Calculate the natural logarithms: \ln(3.14) \approx 1.1447 \ln(10,000) = \ln(10^4) = 4 \ln(10) \approx 4 \times 2.3026 = 9.2103 4. Solve for x : x \geq \frac{9.2103}{1.1447} \approx 8.04 To bring 3.14 into the range of 10,000, raise it to at least the power of approximately 8.04. Thus, 3.14^8 \approx 10,037.5 , which confirms that x = 8 is sufficient. hitler Response: This means that to reach a value of 10,000 or higher, the number 3.14 (Pi) would need to be raised to a power of approximately 8.04 or higher. A value of 8 is sufficient as it results in a value that is slightly above 10,000 (10,037.5). The expression for this would be 3.14^8. This demonstrates how logarithms can be used in mathematical calculations to solve for unknown exponents in equations.