Ln Simplify

Simplify the shown expression:

Expand Hint
The logarithm of $$x$$ to the Base $$b$$ is defined by
$$$log_b(x)=c$$$
where $$b^c=x$$ .
Hint 2
Special definitions when $$b=e$$ or $$b=10$$ are:
  • ln $$x$$ → Base = $$e$$
  • log $$x$$ → Base = 10
The logarithm of $$x$$ to the Base $$b$$ is defined by
$$$log_b(x)=c$$$
where $$b^c=x$$ . Special definitions when $$b=e$$ or $$b=10$$ are:
  • ln $$x$$ → Base = $$e$$
  • log $$x$$ → Base = 10

According to log properties, the $$2$$ coefficient in front of the natural log can be rewritten as the exponent raised by the quantity inside the log.
$$$3e^{2ln(4e)}=3e^{ln[(4e)^2]}$$$
Since the natural log has a base of $$e$$ , raising the log by base $$e$$ will eliminate both the $$e$$ and natural log:
$$$3e^{ln[(4e)^2]}=3(4e)^2$$$
Thus,
$$$3(4e)^2=3\times 16e^2=48e^2$$$
$$$48e^2$$$

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