We want to solve the
given exponential equation
and give the answer in exact form,
as well as round it to
three decimal places.
So the given exponential equation
is 10 raised to the power
of negative 2x equals 12.4.
Because the exponential
part is already isolated,
there are two main ways
to solve this equation.
We could take the common log
on both sides of the equation
because we have base 10 here,
or we could write the exponential equation
as the equivalent log equation
using the definition of
a logarithm, shown here.
For this example, we'll be using
the definition of a logarithm.
Where this log equation and
this exponential equation
are equivalent, where b is the base,
a is the exponent,
and n is the number.
So let's write this exponential
equation as a log equation.
So we know we'll have a logarithm.
And because we have an equation
we'll have an equal sign.
Let's first identify the base.
Well the base is 10 so
we have a log base 10,
which is common log so
we can actually leave
the base of 10 off in the next step
because if no log is given, it is base 10.
Next, a logarithm is an
exponent and therefore,
the logarithm is equal to negative 2x.
Again, notice the exponent is a here
and the logarithm is equal
to a in the log equation.
And then finally, because
the exponential part
is equal to the number 12.4,
our log equation is log base 10
of 12.4 equals negative 2x.
And just to double check
that we have this correct,
if you were to write the log equation
as the exponential equation,
we'd start with the base and
work around the equals sign
to form the exponential equation.
10 raised to power of negative 2x
equals 12.4 so our log
equation is correct.
And again, because we have log base 10
this is common log so if we
want to we can rewrite this
as just log of 12.4
equals negative 2x.
Notice in this form, we
can easily solve for x
by dividing both sides by negative two.
So simplifying, we have x equals
We can leave this as a quotient
or we can also write this
as negative 1/2
times common log 12.4.
So this is our exact solution
that we enter into our first cell.
Again we have x equals negative 1/2
times the common log of 12.4.
Now let's also get our
decimal approximation
using the calculator.
We have negative 1/2.
And common log is this button here.
Again, common log is log base 10.
We have common log 12.4,
close parentheses and enter.
To three decimal places,
our solution is
approximately negative 0.547.
Let's verify that our solution
does satisfy the equation.
So begging with the original
exponential equation,
let's substitute the exact
value of x into the equation,
which would give us 10 raised
to the power of negative two
times negative 1/2
times the common log of 12.4
equals 12.4.
Simplifying the exponent
is how we have negative two
times negative 1/2 which is positive one,
so we can write the left side as 10
raised to the power of common log 12.4
equals 12.4.
Now we should recognize
here this is a property
of logarithms where if we have 10
raised to the power of
log base 10 of 12.4,
this simplifies perfectly to 12.4.
So if the base matches the base
of the logarithm in the exponent,
it simplifies perfectly to the number part
of the logarithm in the exponent.
Let's just say we didn't
recognize this, though,
let's also verify the left half
of our equation here does equal 12.4.
So enter 10 raised to the power of
negative two times negative 1/2,
common log 12.4,
close parentheses for the common log,
another close parentheses
for the product, and enter.
And notice how this does
verify our solution is correct.
I hope you found this helpful.
