Basic Analysis I — Real Analysis · Problem 5.4.2

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · logarithm-bases

Problem

Let b>0b > 0, b≠1b \neq 1 be given. a) Show that for every y>0y > 0, there exists a unique number xx such that y=bxy = b^x. Define the logarithm base bb, log⁡b:(0,∞)→R\log_b : (0,\infty) \to \mathbb{R}, by log⁡b(y):=x\log_b(y) := x. b) Show that log⁡b(x)=ln⁡(x)ln⁡(b)\log_b(x) = \dfrac{\ln(x)}{\ln(b)}. c) Prove that if c>0c > 0, c≠1c \neq 1, then log⁡b(x)=log⁡c(x)log⁡c(b)\log_b(x) = \dfrac{\log_c(x)}{\log_c(b)}. d) Prove log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y), and log⁡b(xy)=ylog⁡b(x)\log_b(x^y) = y\log_b(x).

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