# 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.

https://brainmass.com/math/functional-analysis/mathematics-functional-analysis-213008

#### 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.

Logarithm function -

Deliverable Length: 2-4 paragraphs

Details: Many different kinds of data can be modeled or measured easily using exponential and logarithmic functions. For example, consider ideal gases: Pressure versus volume give a curved plot, but a log graph of pressure versus volume provides a linear plot that is easier to interpret. Log plots are also used in the study of the rates of reactions; for many reactions, the log of remaining reactant concentration versus time is linear.

For this Discussion Board, use the Library and other resources to find at least two examples of scientific data that are measured and reported using log scales. Examples can be found in such fields as chemistry, astronomy, and geology.

Deliverable Length: 2-4 paragraphs

Details: Many different kinds of data can be modeled or measured easily using exponential and logarithmic functions. For example, consider ideal gases: Pressure versus volume give a curved plot, but a log graph of pressure versus volume provides a linear plot that is easier to interpret. Log plots are also used in the study of the rates of reactions; for many reactions, the log of remaining reactant concentration versus time is linear.

For this Discussion Board, use the Library and other resources to find at least two examples of scientific data that are measured and reported using log scales. Examples can be found in such fields as chemistry, astronomy, and geology.

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