We are asked to evaluate
the given expressions.
We will evaluate both
expressions using the
inverse property of logarithms
and exponentials shown here.
Both of these properties exist because
logarithms and exponentials
are inverses of one another,
and therefore undo each other.
So first we have log base b of b
raised to the power of x equals x,
and b raised to the power of
log base b of x also equals x.
Notice in both cases the
base of the exponential is b
and the base of the log is also b.
For number one we have log 1,000,
notice how the base of
the log is not given,
which means this is common
log, or log base 10.
So log 1,000 equals log base 10 of 1,000,
so if we can write 1,000 with base 10,
we can use this first property
to simplify the expression.
Well 1,000 is equal to
10 times 10 times 10,
which equals 10 raised to the
power of three or 10 cubed,
which means log base 10 of 1,000 equals
log base 10 of 10 cubed,
which fits the form of the first property,
and therefore is equal to three.
Number two we have e raised to the
power of natural log five.
Natural log is log base e,
and therefore e raised to
the power of natural log five
equals e raised to the
power of log base e of five,
notice how this fits the
form of the second property,
where we have b raised to the power of
log base b of x equals x, so
because we have base e here,
and it's raised to the
power of log base e,
this simplifies to five.
I hope you found this helpful.
