Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 1 - Section 1.5 - Inverse Functions and Logarithms - 1.5 Exercises: 58


(a) $$e^{\ln 300} = 300,\quad \ln(e^{300}) = 300.$$ (b) It will give you the correct value for $e^{\ln 300}$ but it won't be able to calculate $\ln(e^{300})$ because $e^{300}$ is a huge number that cannot be handled by the calculator's memory.

Work Step by Step

(a) Since $\ln$ and natural exponential function are inverse to each other then their composition will be equal to its' argument because $$f(f^{-1}(x))=f^{-1}(f(x)) = x$$ so we have $$e^{\ln 300} = 300,\quad \ln(e^{300}) = 300.$$ (b) If you use your calculator if you wont be able to evaluate $\ln(e^{300})$ because it will first try to calculate $e^{300}$ and this is a huge number of the order of $10^{150}$ and there is not enough memory in your calculator to store it. If you try to evaluate $e^{\ln 300}$ you will get the correct value because the calculator would first find what's $\ln 300$, which is around $5.7$, and when it exponentiates this it will give you $300$.
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