Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.4 Derivatives of Logarithmic Functions - 6.4 Exercises: 1

Answer

The natural logarithms is written as $y = ln(x)$ i.e., loge x and logarithms to base 10 is shown as $y = log_{10} x$. The main reason is that Logarithmic functions are the inverse of exponential functions. The inverse of the exponential function is given as $y = log_{e} x$ and inverse of this $x = e^{y}$ ; this function is special for many reasons. One of the first approaching thing we notice about this function is that its derivative $ \frac {dy} {dx} = e^{y}$ has its own derivative and exists in natural growth and decay in physical world. Log with base e functions are more frequently used because of finding exact solutions of differential and partial differential. We usually use the variable t rather than x as our independent variable, because our growth and decay occur over an interval of time. Log with base 10 is commonly used to simplify manual calculations and it is also related with decimal system, if we calculate log of a number with base 10, then integer just greater than that calculate value. The relation between natural logarithms and base 10 logarithms is $ln x = 2.303 log_{10}x$

Work Step by Step

The natural logarithms is written as $y = ln(x)$ i.e., loge x and logarithms to base 10 is shown as $y = log_{10} x$. The main reason is that Logarithmic functions are the inverse of exponential functions. The inverse of the exponential function is given as $y = log_{e} x$ and inverse of this $x = e^{y}$ ; this function is special for many reasons. One of the first approaching thing we notice about this function is that its derivative $ \frac {dy} {dx} = e^{y}$ has its own derivative and exists in natural growth and decay in physical world. Log with base e functions are more frequently used because of finding exact solutions of differential and partial differential. We usually use the variable t rather than x as our independent variable, because our growth and decay occur over an interval of time. Log with base 10 is commonly used to simplify manual calculations and it is also related with decimal system, if we calculate log of a number with base 10, then integer just greater than that calculate value. The relation between natural logarithms and base 10 logarithms is $ln x = 2.303 log_{10}x$
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