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Mathematics - Functional Analysis

Please see the attached file for the fully formatted problems.

1. If y = c^x express x in terms of y. (c is a positive constant)

2. a) Sketch the graph of y = 5^x for ?2 < x < 3, clearly showing the
key features of this exponential function.
(b) On the same graph, sketch the function y = log5 x. Explain the relationship between the two functions.

3. If 8 log6 x + log6 y = 2, solve for y in terms of x.

4 Show that log3 () may be written as log3 2 + 2(log3 a ? log3 5).

5 If 4^(2x-3) = 7, find a value for x.

6. The intensity, I, of an earthquake is related to a number, R, on the Richter scale according to the equation R = log I ? log I0, where Io is a minimum level for comparison. Solve for I in terms of R and I0.
7. The bacterial population in a certain culture is given by N = N0 (1 .6)t where t is in hours and N0 is the initial population (when t = 0). Determine how long it takes for this population to double in size.

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Solution Preview

(1) ln y = x * ln c

x = ln y / ln c = ln y (to the base c)

(2) The graph is attached (Blue graph = 5^x and Red graph = log x (to the base 5).

The two functions are inverses of each ...

Solution Summary

A Complete, Neat and Step-by-step Solution is provided. Graph also provided in the attached file.