Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 7: Transcendental Functions - Section 7.3 - Exponential Functions - Exercises 7.3 - Page 391: 79

Answer

$$\frac{dy}{dt} =\frac{1}{t}\left(\log _{2} 3\right) 3^{\log _{2}t}$$

Work Step by Step

Given $$ y = 3^{\log _{2} t}$$ Since $$\log_{a}z=\frac{\ln z}{\ln a}$$ So, we have \begin{aligned} y& = 3^{\log _{2} t}\\ &=3^{\ln t /\ln 2}\\ & \Rightarrow \frac{dy}{dt}=\left[3^{\ln t/ln2}(\ln 3)\right]\left(\frac{1}{t \ln 2}\right)\\ &\ \ \ \ \ \ \ \ \ \ \ \ =\left[3^{\ln t/ln2} \right]\left(\frac{\ln 3}{t \ln 2}\right)\\ &\ \ \ \ \ \ \ \ \ \ \ \ =\frac{1}{t}\left(\log _{2} 3\right) 3^{\log _{2}t} \end{aligned}
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