Answer the following questions using concepts from logarithmic and exponential functions. (1) A reti ...

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Answer the following questions using concepts from logarithmic and exponential functions. (1) A retirement account is opened with an initial deposit of $8,500 and earns 8.12% interest compounded monthly. What will the account be worth in 20 years? What if the interest were calculated as simple interest? Show the situation in a graph, and say from what point one is better than the other. (2) Graph the function f(x) = 5(0.5)^(-x) and its reflection about the line y = x on the same axes, and give the x-intercept of the reflection. Prove that a^x = e^(x ln a). (3) How long will it take before twenty percent of a 1,000-gram sample of uranium-235 has decayed? The decay equation is A(t) = A_0 e^(Kt), where t is the time for the decay and K is characteristic of the material. Suppose T is the half-life, the time it takes for half of the unstable material to decay. Prove that K = ln(0.5)/T. What is K for uranium-235 (T = 703,800,000 years)? Show the steps of your reasoning.