a. Use the definition of logarithm to show that logbbx = x for all real numbers x. b
Chapter 11, Problem 10(choose chapter or problem)
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 25 in Section 7.3, if \(f: X \rightarrow Y\) and \(g: Y \rightarrow X\) 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.
Text Transcription:
log _b b^x=x
b^log _b x=x
f: X \rightarrow Y
g: Y \rightarrow X
g \circ f=I_X
f \circ g=
I_Y
log _b
exp _b
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