- Natural Logarithms.
Definition of Natural
Logarithms.
A logarithm with a special base
called e is a natural logarithm
where e is an irrational number.
Rounded to the nearest
10 thousandth, E = 2.7183.
Definition, Natural Logarithm.
A natural logarithm
is a logarithm with base e.
We write ln[A]
to represent logE[A].
Exponential, Natural
Logarithmic Forms Property.
For A greater than 0,
the equations, ln[A] = C
and A to the C power = A
are equivalent.
The natural log of A
is the exponent on the base E
that gives A.
For this example,
use a calculator to find ln[40].
On our calculator, we can find
our natural log right here.
If you look above it,
you have your E to the X power.
To find ln[40], press your
ln button and then type 40.
We don't have to close
the parenthesis
since we're not doing
anything else,
but we can press enter,
and we get 3.688879454.
Using four decimal places,
ln[40] = 3.6889.
In this example,
find the natural logarithm.
Use our calculator
to verify the result.
For part a, we have ln[E]
to the 4th power.
Since the natural log
is the same as logE.
We know from our earlier
sections
that if your base
and your answer are the same,
they cancel out living you
with just the power.
So, this result should be 4.
Going to our calculator,
we type our natural log,
then in the parenthesis, we want
to type E to the 4th power.
So, second, natural log,
it already gives us the exponent
when we use E, 4,
then we close parenthesis,
enter, and I do get 4.
So ln[E] to the 4th power = 4.
For part B, we have 1/2 ln[E]
to the 6th power.
Again, the natural log and E
cancel each other out
living me with just 6,
so that means we have 1/2 x 6,
which gives us 3.
Calculating.
We have parenthesis 1
divided by 2 x ln.
Second, ln, to get E
to the power of 6, and we get 3,
so 1/2 x ln[E] to the 6 = 3.
In this example,
solve the equation.
Round any solutions
to the fourth decimal place.
2 x ln[5X] - 3 = 1.
We'll work this problem
similar to working our problems
with log,
and we have to solve
for a variable.
It's helpful to remember that
ln[A] = C,
and E to the C power = A.
Our first step is to isolate
ln[5X].
I can do that by adding 3
to both sides,
living me with 2 x ln[5X] = 4,
then dividing both sides by 2,
ln[5X] = 2.
Now we're ready to rewrite it
in the exponential form.
So, I have E to the 2nd power
= 5X, dividing both sides by 5,
I have E squared divided by 5
= X,
and now I'm ready to calculate.
So, to get my E, 2nd, ln, 2
divided by 5, enter,
and I get 1.47781122.
Rounding it out to four decimal
places, X = 1.4778.
In this example, we want
to solve the equation,
round any solutions
to the fourth decimal place.
7 x E to the power
of 2P + 10 = 100.
For this problem, we need
to rewrite it as a natural log.
Our first step is to isolate
my E to the C,
I have it equal to A, so that we
can rewrite it as ln[A] = C.
Dividing both sides by 7,
I have E to the power
of 2P + 10 = 100 divided by 7.
Now, I can rewrite it
as a natural log.
My ln[A], which in this case
is 100 divided by 7 = C,
which is 2P + 10.
Calculating.
I have ln[100] divided by 7,
and that gives us 2.659260037.
It's best to have as exact
an answer as possible.
So, for best results,
you want to wait until you're
done calculating to round.
With the TI calculator,
that can actually help us
because in all of our
calculations,
we can continue to use
this answer.
For my work, I'm going to round
it to four decimal places.
2.6593 = 2P + 10.
Now, I want to subtract 10
from both sides.
Calculating.
I can just type minus 10,
it will immediately subtract it
from the answer, enter,
and I have -7.340739963.
I'm still going to round
to the fourth decimal place,
which gives me -7.3407 = 2P.
I want to divide both sides
by 2.
Going back to my calculator,
I can just press my division
sign, 2, enter,
and I get -3.670369982.
Rounding this to the fourth
decimal place,
I have -3.6704 = P.
So, for my solution,
so for my solution, P = -3.6704.
Natural Logarithms.
Our natural logarithm
is a logarithm
with a special base called E,
where E is an irrational
number, 2.71821828.
We can write the natural
logarithm as ln[A],
where LN represents logE.
For all logarithms,
assume that X greater than 0,
Y greater than 0, B greater
than 0, and b does not equal 1.
The following properties
will apply.
For the property of a logarithm,
logB[1] = 0.
For the natural logarithm,
ln[1] = 0.
Ln[1] is 0.
For the property of logarithm,
logB[B] = 1.
For natural logarithm,
ln[E = 1] because ln[E] is 1.
For a logarithm, logB[B] to
the X = X for any real number X.
For your natural logarithm,
ln [E to the X] = X.
for real number x composing
y = lnx with Y = E to the E.
For logarithms,
b to the exponent logB[X] = X.
For natural logarithm,
e with an exponent of ln[X] = X
composing Y = E to the X with Y
= ln[X].
For logarithm, logB[X to the P]
= P x logB[X].
For a natural logarithm,
ln[X to the P] = P x ln[X],
which is your power property.
For logarithm logB[X] + logB[Y]
= logB[X x Y].
For natural logarithm,
ln[X] + ln[Y] = ln[X x Y],
which is your product property.
For logarithm logB[X] - logB[Y]
= logB[X divided by Y].
For natural logarithm, ln[X]
- ln[Y] = ln[X divided by Y]
which is your quotient property.
