Question: a. Use the definition of logarithm to show that \(\log _{b} b^{x}=x\) for all real numbers \(x\).

b. Use the definition of logarithm to show that \(b^{\log _{b} x}=x\) for all positive real numbers \(x\).

c. By the result of exercise 26 in Section 7.4, if \(f: X \rightarrow Y\) and \(g:Y\to X\) are functions and \(g \circ f=i_{X}\) and \(f{}^\circ g={{i}_{Y}}\)

then \(f\) and \(g\) are inverse functions. Use this result to show that \(\log _{b}\) and \(\exp _{b}\) (the exponential function with base b) are inverse functions.

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