### Introduction to differentiation: Derivatives of exponential functions and logarithms

### Rules of calculation for exponential functions and logarithm

Before we proceed with differentiation, we list the rules of calculation that are most frequently used for the natural exponential function and logarithm.

Rules of calculation for the natural exponential function and logarithm

Let #x#, #y#, #a# be positive numbers with #a\ne1# and let #u# and #v# be arbitrary numbers.

- #\ln(1) = 0#
- #\ln(x\cdot y) =\ln(x)+\ln(y)#
- #\ln\left(\dfrac{1}{x}\right) =-\ln(x)#
- #\ln\left(\dfrac{x}{y}\right) =\ln(x)-\ln(y)#
- #\ln\left(x^u\right) =u\cdot \ln(x)#
- #\log_a(x) =\dfrac{\ln(x)}{\ln(a)}#
- #\e^{\ln(x)}=x#
- #\ln\left(\e^u\right) = u#
- if #u\lt v#, then #\e^u\lt \e^v#; in other words: #\exp# is increasing on #\mathbb{R}#
- if #x\lt y#, then #\ln(x)\lt \ln(y)#; in other words: #\ln# is increasing on #\ivoo{0}{\infty}#

All statements follow directly from *Properties of the logarithm *and known rules of arithmetic.

Rule 6 makes it possible to express the logarithm to the base #a# in terms of the natural logarithm.

Simplify the expression #\exp(\ln\left( x^{4} \right) - 9\cdot\ln\left( y \right))#.

Solution #{{x^4}\over{y^9}}#

This can be seen as follows:

\[ \begin{array}{rclcl}

\e^{\ln{(x^4)}-9 \cdot \ln{(y)}} &=&\e^{\ln{(x^4)}-\ln{(y^9)}} &\phantom{xx}&\color{blue}{a\cdot\ln(b)=\ln\left(b^a\right)}\\

&=& \frac{\e^{\ln{(x^4)}}}{\e^{\ln{(y^9)}}} &\phantom{xx}&\color{blue}{\e^{a-b}=\frac{\e^a}{\e^b}}\\

&=& \frac{x^4}{y^9}&\phantom{xx}&\color{blue}{\e^{\ln\left(a\right)}=a}\\

\end{array} \]

This can be seen as follows:

\[ \begin{array}{rclcl}

\e^{\ln{(x^4)}-9 \cdot \ln{(y)}} &=&\e^{\ln{(x^4)}-\ln{(y^9)}} &\phantom{xx}&\color{blue}{a\cdot\ln(b)=\ln\left(b^a\right)}\\

&=& \frac{\e^{\ln{(x^4)}}}{\e^{\ln{(y^9)}}} &\phantom{xx}&\color{blue}{\e^{a-b}=\frac{\e^a}{\e^b}}\\

&=& \frac{x^4}{y^9}&\phantom{xx}&\color{blue}{\e^{\ln\left(a\right)}=a}\\

\end{array} \]

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