Logarithmic Functions (Rules) and Derivatives of Logarithmic Functions (More Rules)

8 Basic Properties of Logarithmic Functions:

1. $log_b1=0$

2. $log_bb=1$

3. $log_bb^x=x$

4. $b^{log_bx}=x, x>0$

5. $log_bMN=log_bM+log_bN$

6. $log_b\frac{m}{n}=log_bM-log_bN$

7. $log_bM^p=plog_bM$

8. $log_bM=log_bN$ if and only if $M=N$

7 More Rules/Quick Equations for Derivatives of Logs and Natural Logs:

1. $D_x(e^x)=e^x$ so basically the derivative of $e^x$ is $e^x$ (examples to come)

2. $b^x=e^{xlnb}$ (for example $y=5^x$ is also equal to $y=e^{xln5}$ which you can then solve for using the chain rule) OR you can use the following equation instead:

3. For a function $y=b^x$ , $\dfrac{dy}{dx}=b^x*lnb$

4. $e^{lnx}=x$ as well as $ln(e^x)=x$ (this is not a derivative equation but is helpful in finding the derivative).

5. $D_xlnx=\frac{1}{x}$ the derivative of lnx is always $\frac{1}{x}$

6. Change of Base Formula (also not a derivative equation but helpful) $log_ba=\dfrac{log_ca}{log_cb}$ aka $log_ba=\dfrac{lna}{lnb}$ for any $c>0.$ and c≠1.

7. For any valid log base b (meaning b>0, and b≠1): when $y=log_bx$ , $\frac{dy}{dx}=\dfrac{1}{xlnb}$

Examples using these equations in order:

A.Using $D_x(e^x)=e^x$ I can show you how to use this information with the previous rules (chain, quotient, product, constant multiple) to get the derivative using e.

1. Using $D_x(e^x)=e^x$ with the constant multiple rule:

$y=7e^x$

$y'=7e^x$ using the constant multiple rule

2. using $D_x(e^x)=e^x$ with the product rule:

$y=xe^x$

$y'=xe^x+e^x$

or in real numbers

$y=x^2e^x$

$y'=x^2*e^x+e^x*2x$

$y'=x^2e^x+2xe^x$

$y'=e^xx(x+2)$ simplified.

4. using $D_x(e^x)=e^x$ with the quotient rule:

$y=\frac{x}{e^x}$

$y'=\dfrac{e^x*1-x*e^x}{(e^x)^2}$

$y'=\dfrac{e^x-xe^x}{(e^x)^2}$

$y'=\dfrac{1-x}{e^x}$ simplified.

5. using $D_x(e^x)=e^x$ with the chain rule:

$g(x)=e^{9x}$

$g'(x)=e^{9x}*9$

$g'(x)=9e^{9x}$

B. Using Rule #3: For a function $y=b^x$ , $\dfrac{dy}{dx}=b^x*lnb$ here is an example:

$y=11^{x^5+9}$

$y'=11^{x^5+9}*ln11$

C. Using the rule $\frac{d}{dx}lnx=\frac{1}{x}$ we can use the following example:

$y=ln(2x+5)$

$y'=\dfrac{1}{2x+5}*2$ (remember we also have to incorporate the chain rule since it’s a function within a function)

$y'=\dfrac{2}{2x+5}$

If you want to get a little more complicated with it, and remember some of the logarithmic rules, you can solve for even more difficult equations. Let’s take the rule $lnM^p=plnM$ for our first example:

$y=ln(x^5)$

$y=5lnx$ using that rule above

$y'=5*\frac{1}{x}$

$y'=frac{5}{x}$

Another useful rule we went over $ln\frac{M}{N}=lnM-lnN$

$y=ln(\dfrac{\sqrt{x}}{8x^2-3})$

$y=ln\sqrt{x}-ln(8x^2-3)$

$y=lnx^{\frac{1}{2}}-ln(8x^2-3)$

$y=\frac{1}{2}lnx-ln(8x^2-3)$

$y'=\frac{1}{2x}-\frac{1}{8x^2-3}*16x$

$y'=\frac{1}{2x}-\frac{16x}{8x^2-3}$

TADA!!!!

Finally, I’ll show you how you can use the rule $lnMN=lnM+lnN$

Example:

$y=ln(x^4(3x+7)^8)$

$y=lnx^4+ln(3x+7)^8$

$y=4lnx+8ln(3x+7)$

$y'=\frac{4}{x}+\dfrac{8}{3x+7}$

D. Finally I’ll use the rule $y=logb_bx$ $y'=\dfrac{1}{xlnb}$

Example:

$y=4log_6x$

$y'=\dfrac{4}{xln6}$

Hopefully this was helpful to you. I want to make a video soon, but I have not yet found a decent way to work the camera while doing math. I’m working on that. I found today’s homework challenging. The last few problems took me a few tries, but I think these rules really help and if I had taken the time to bust them out for the homework I may have gotten finished faster. Meh. Next time.

Hope you all have a fantastic weekend! I found this on Pinterest and thought it was amusing.