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Start with the given equation.
Rewrite the left side using the identity
Simplify
Multiply EVERY term by the LCD 9 to clear out the fractions.
Simplify
Rewrite 9 as
Multiply the exponents.
Factor out the exponent 2 using the identity
Now let's make a substitution to make things easier on us. Let
(since there are 2 copies of
)
Replace each
with 'z'
Notice that the quadratic
is in the form of
where
,
, and
Let's use the quadratic formula to solve for "z":
Start with the quadratic formula
Plug in
,
, and
Negate
to get
.
Square
to get
.
Multiply
to get
Rewrite
as
Add
to
to get
Multiply
and
to get
.
Take the square root of
to get
.
or
Break up the expression.
or
Combine like terms.
or
Simplify.
So the solutions are
or
Recall that we let
. Since
for all 'x', the solution
isn't possible (since
)
So we're only going to focus on
Start with the given equation.
Plug in
Convert to logarithmic form.
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Answer:
So the only solution is
(