## Ln Simplify

Simplify the shown expression:

##
__
__**Expand Hint**

**Expand Hint**

The logarithm of
$$x$$
to the Base
$$b$$
is defined by

$$$log_b(x)=c$$$

where
$$b^c=x$$
.

##
__
__**Hint 2**

**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,

$$$3(4e)^2=3\times 16e^2=48e^2$$$

$$$48e^2$$$

Similar Problems from FE Section:

**Logarithms**

434. Logarithm

461. Log

465. Base Log

468. Ln Base

470. Natural Log