We want to solve the exponential equation
and we're asked to give
the answer in exact form,
as well as rounded to
three decimal places.
So we want to solve the
exponential equation
e raised to the power of
negative seven x equals 11.3.
Because the exponential
part is already isolated,
we can solve this by
taking the natural log
of both sides of the equation,
or by applying the
definition of a logarithm
and write the equivalent log equation.
For this example,
we'll be applying the definition
of a logarithm shown here.
These two equations are equivalent,
and we'll use this definition
to write the exponential
equation as a log equation
where b is the base, a is the exponent,
and n is the number.
Writing the equivalent log equation,
we know we'll have a logarithm.
We'll also have an equals sign.
Let's begin by identifying the base.
The base is e, so we have log base e.
Which is actually natural log,
so we'll change the
notation in the next step.
A logarithm is an exponent,
the exponent is negative seven x,
so the logarithm is equal
to negative seven x.
And the exponential is
equal to the number 11.3,
and therefore we have log base e of 11.3
equals negative seven x.
And for one last check,
if you were to go from
the log equation to the
exponential equation,
we'd start with the base and
work around the equals sign.
We would have e raised to the power
of negative seven x equals 11.3.
So our log equation is correct,
let's go ahead and replace
log base e with natural log,
so our equation is natural log
11.3 equals negative seven x.
And now we can easily solve for x
by dividing both sides by negative seven.
Simplifying, we have x equals.
We can leave this as a quotient
or we could also write
this as negative 1/7
natural log 11.3.
Dividing by negative seven is the same
as multiplying by negative 1/7.
So our exact solution is
x equals negative 1/7 natural log 11.3.
So this is what we enter
into our first answer cell.
And now let's round this
value to three decimal places.
So using the calculator,
we'll enter negative one divided
by seven natural log 11.3.
To three decimal places,
our solution is
approximately negative 0.346.
Let's verify our solution is correct
by substituting this
value into the equation
and make sure it satisfies the equation.
So beginning with e raised to the power
of negative seven x equals 11.3.
We'll perform the substitution for x,
so we would have e raised to
the power of negative seven
times negative 1/7 natural
log 11.3 equals 11.3.
Looking at the exponent,
notice how we have negative
7 times negative 1/7
which is one,
so this would be e raised
to the power of natural log
of 11.3 equals 11.3.
Now if we know our
properties of logarithms,
we should recognize that on the left side,
because we have base e here,
raised to the power of
log base e or natural log.
This simplifies perfectly to 11.3.
So this verifies our solution is correct,
but let's also verify
it on the calculator.
Let's enter the left
side of the equation here
to make sure this is equal to 11.3.
So if we press second, natural log,
that brings up e raised to the power of,
and we enter negative seven.
And then in parentheses
we have negative one divided
by seven natural log 11.3.
Closed parenthesis for the natural log,
and closed parenthesis for the product.
And enter.
And notice how we do get 11.3,
verifying our solution is correct.
I hope you found this helpful.
