Eureka! I’ve got it! (Logs, Natural Logs, and e)

I finally got it. I had to go back to MathXL and do the optional homework on logs and natural logs, but it finally started making sense. I want to post the log rules I hadn’t found, but desperately need to know:

1. $ln(xy) = lnx + lny$

Example: $ln8 + lny = ln(8y)$

2. $ln(\frac{x}{y})=ln(x)-ln(y)$

Example: $ln4-2ln4+ln8=ln(\dfrac{4*8}{4^2})$ which simplifies to $ln(2)$

3. $ln(\frac{1}{x})=-lnx$

4. $ln(x^b)=blnx$

Example: $\frac{1}{2}lnx = lnx^{\frac{1}{2}}$

5. $log_bx$ can be re-written as $b^y=x$

Example: $log_{22}x=-3$ can be written as $22^{-3}=x$

6. $e^{lnx+lny}=xy$ because $e^{lnx} = x$

7. $b^{xy}=(b^x)^y$

8. $b^{x+y}=b^x*b^y$

9. Using those rules for e – $e^{lnx} = x$ so $e^{12lnx}=(e^{lnx})^{12}=x^{12}$

10. Another example where you can use those rules with e: $e^{lnx^3+3lnz}=x^3z^3$

I also was having a bit of a hard time dealing with exponential x’s with our 2nd derivatives and found this to be useful:

IF you end up with a problem that looks like this: $log_{81}27=x$ can be translated into $81^x=27$ how do we solve for the exponent? Factor the 81 and 27 like so: $(3^4)^x=3^3$ then solve for the exponents $4x=3$ $x=\frac{3}{4}$

All these should be useful in helping us with second derivatives and remembering natural log rules, hopefully.

I’m excited to be having a study group with the girls I sit next to in Calculus! I think it will be really useful to hash out all the details, and hopefully do really well on Exam #2 so we can enjoy our Spring Break!

Finally, I leave you with this: