What is Natural Logarithm

1) Evaluate without using a calculator: $3 \ ln \ e$

$\color{red}{ln \ e \ = \ log_e \ e \ = \ 1 \ ⇒ \ 3 \ ln \ e \ = \ 3}$

2) Evaluate without using a calculator: $-3 \ ln \ e^5$

$\color{red}{ln \ e \ = \ log_e \ e \ = \ 1 \ ⇒ \ -3 \ ln \ e^5 \ = \ -3 \times 5 \ ln \ e \ = \ -15}$

3) Evaluate without using a calculator: $9 \ ln \ \frac{1}{e}$

$\color{red}{ln \ e \ = \ log_e \ e \ = \ 1 \ ⇒ \ 9 \ ln \ \frac{1}{e} \ = \ -9 \ ln \ e \ = \ -9}$

4) Evaluate without using a calculator: $6 \ ln \ \frac{1}{e^5}$

$\color{red}{ln \ e \ = \ log_e \ e \ = \ 1 \ ⇒ \ 6 \ ln \ \frac{1}{e^5} \ = \ -6 \ ln \ e^5 \ = \ -6 \times 5 \ ln \ e \ = \ -30}$

5) Evaluate without using a calculator: $e^{ln \ \frac{1}{e}}$

$\color{red}{e^{ln \ \frac{1}{e}} \ = \ \frac{1}{e}}$

6) Solve for $x$: $e^x \ = \ 8$

$\color{red}{e^x \ = \ 8 \ ⇒ \ ln \ e^x \ = \ ln \ 8 \ ⇒ \ x \ ln \ e \ = \ ln \ 8 \ ⇒ \ x \ = \ \frac{ln \ 8}{ln \ e} \ = \ ln \ 8}$

7) Solve for $x$: $ln \ x \ = \ 11$

$\color{red}{ln \ x \ = \ 11 \ ⇒ \ e^{ln \ x} \ = \ e^{11} \ ⇒ \ x \ = \ e^{11}}$

8) Solve for $x$: $ln \ (x \ + \ 5) \ = \ 20$

$\color{red}{ln \ (x \ + \ 5) \ = \ 20 \ ⇒ \ e^{ln \ (x \ + \ 5)} \ = \ e^{20} \ ⇒ \ x \ + \ 5 \ = \ e^{20} \ ⇒ \ x \ = \ e^{20} \ - \ 5}$

9) Solve for $x$: $ln \ (3x \ - \ 9) \ = \ 17$

$\color{red}{ln \ (3x \ - \ 9) \ = \ 17 \ ⇒ \ e^{ln \ (3x \ - \ 9)} \ = \ e^{17} \ ⇒ \ 3x \ - \ 9 \ = \ e^{17} \ ⇒ \ 3x \ = \ e^{17} \ + \ 9 \ ⇒}$ $\color{red}{x \ = \ \frac{e^{17} \  +\ 9}{3} \ = \ x \ = \ \frac{1}{3} \ e^{17} \ + \ 3}$

10) Solve for $x$: $ln \ x \ = \ 2 \ ln \ 5 \ + \ 2 \ ln \ 2$

$\color{red}{ln \ x \ = \ 2 \ ln \ 5 \ + \ 2 \ ln \ 2 \ ⇒ \ e^{ln \ x} \ = \ e^{2 \ ln \ 5 \ + \ 2 \ ln \ 2} \ ⇒ \ e^{ln \ x} \ = \ e^{ln \ 5^2 \times 2^2} \ ⇒ }$$\color{red}{ \ x \ = \ 5^2 \times 2^2 \ = \ 100}$