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Start with the given equation.
Use the change of base formula.
Pull down the exponents using the identity
Pull out the constants.
Factor out the GCF
Multiply both sides by 2.
. Take note that
. In other words,
Make the proper substitutions.
Multiply EVERY term by the LCD 'z' to clear out the fraction.
Subtract 2z from both sides.
Notice that the quadratic
is in the form of
Let's use the quadratic formula to solve for "z":
Start with the quadratic formula
Take the square root of
Break up the expression.
Combine like terms.
So the only solution in terms of z is
But remember, we let
Go back to that previous equation.
Multiply both sides by
Since the bases are equal, the arguments are equal.
So the solution is