Find the value of 𝑥 in the
equation three to the power of two 𝑥 minus one equals one over 81.
The equation we’ve been given is an
exponential equation, where the unknown 𝑥 appears in the exponent. To solve such equations, we usually
need to express both sides as powers of the same base.
We should recall that as well as
being equal to nine squared, 81 is also equal to three to the fourth power. This is helpful because three is
the base on the other side of the equation. We can therefore rewrite the
equation as three to the power of two 𝑥 minus one is equal to one over three to the
We now have both sides in terms of
the same base, but the power term is in the denominator on the right-hand side. We can recall the reciprocal law of
exponents, which states that for a nonzero base 𝑎, one over 𝑎 to the 𝑚th power is
equal to 𝑎 to the negative 𝑚th power. By applying this law, we can
express the right-hand side of the equation as three to the power of negative
Both sides of the equation are now
written as the base three raised to some power. We can now recall a second law of
exponents, which states that if 𝑎 is a real number not equal to negative one, zero,
or positive one and if 𝑎 to the 𝑚th power is equal to 𝑎 to the 𝑛th power, then
𝑚 is equal to 𝑛. In other words, if raising the base
𝑎, which satisfies these conditions to one power, gives the same result as raising
it to another power, then the two powers are equal. We can therefore equate the two
powers to give the equation two 𝑥 minus one is equal to negative four. This is now a linear equation in
the unknown 𝑥 which we can solve in the usual way.
To isolate the 𝑥-term, we add one
to both sides, giving two 𝑥 equals negative three. Then, we divide both sides of the
equation by two to give 𝑥 equals negative three over two. So, by first expressing both sides
of this equation as powers of the same base and then equating the powers, we’ve
found that the value of 𝑥 in the given equation is negative three over two.