ln e^7
Simplify without a calculator

Question

ln e^7
Simplify without a calculator

anonymous · Answer

hi again apparently the answer is 7 but i'd like an explanation as to why if you can help me out.

jdoe0001 · Answer

well   let's see   ok

aum · Answer

$$
\ln(a)^b = b * \ln(a) \
\ln(e)^7 = 7 * \ln(e) \
\ln(e) = 1 \
$$

jdoe0001 · Answer

$\large \bf ln(e^7)\implies log_{\color{red}{ e}}{\color{blue}{ e}}^7=\square \implies {\color{red}{ e}}^\square ={\color{blue}{ e}}^7$

jdoe0001 · Answer

see what $\square$  equals to?

anonymous · Answer

7 im guessing

jdoe0001 · Answer

well, if the base is the same... and the EQUATion is true... then the exponents need to equal each other :)

anonymous · Answer

wait the base is the red e or the blue e?

jdoe0001 · Answer

hmmm   the base is the red one

anonymous · Answer

okay that makes sense also ln is that just logarithm? If so on the calculator which button would i use if it has both "Log"and "Ln"

jdoe0001 · Answer

yeap    is the "natural logarithm"    or $\large ln\iff log_e$

jdoe0001 · Answer

keep in mind the "log cancellation rule" thus
when the log base and the "base of the number inside" match, then $\Large \bf log_{\color{brown}{ a}}{\color{brown}{ a}}^x=x$

anonymous · Answer

Ohhh wow! Cool! Thanks so much!

jdoe0001 · Answer

yw