Question Video: Solving an Exponential Equation Where the Unknown Appears in the Exponent Mathematics • 9th Grade

Find the value of ๐‘ฅ in the equation 3^(2๐‘ฅ โˆ’ 1) = 1/81.

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Video Transcript

Find the value of ๐‘ฅ in the equation three to the power of two ๐‘ฅ minus one equals one over 81.

The equation weโ€™ve been given is an exponential equation, where the unknown ๐‘ฅ 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 ๐‘ฅ minus one is equal to one over three to the fourth power.

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 ๐‘Ž, one over ๐‘Ž to the ๐‘šth power is equal to ๐‘Ž to the negative ๐‘šth power. By applying this law, we can express the right-hand side of the equation as three to the power of negative four.

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 ๐‘Ž is a real number not equal to negative one, zero, or positive one and if ๐‘Ž to the ๐‘šth power is equal to ๐‘Ž to the ๐‘›th power, then ๐‘š is equal to ๐‘›. In other words, if raising the base ๐‘Ž, 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 ๐‘ฅ minus one is equal to negative four. This is now a linear equation in the unknown ๐‘ฅ which we can solve in the usual way.

To isolate the ๐‘ฅ-term, we add one to both sides, giving two ๐‘ฅ equals negative three. Then, we divide both sides of the equation by two to give ๐‘ฅ 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, weโ€™ve found that the value of ๐‘ฅ in the given equation is negative three over two.

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