Solving an exponential equation applying properties of the power

An exponential equation is one in which the unknown variable or variables is/are in the exponent of a power. The exponential equations use basic knowledge of the exponential and logarithmic functions. As such, we will revise them.

To solve them the following properties are used:

  • $$a^0=1$$ for any $$a$$.
  • Two potencies with the samepositivebasis and different from the unit are equal, if, and only if, its exponents are equal. That is to say:$$$2^a=2^b \Leftrightarrow a=b$$$
  • For any $$a \neq 0$$ and $$a\neq 1$$ we have:$$$a^x=b \Rightarrow x= \log_ab$$$

When it comes to solving an exponential equation it can have different forms and, because of that, there are different methods and transformations.

When we have an equation of the type: $$a^{f(x)}=1$$ with $$a\neq 0$$ and $$a\neq 1$$. Then one proceeds with the properties of the power that says that $$f(x)=0$$ since the only exponent that, for any base to give one, is the zero exponent.

$$$10^{x^2+-2}=1 \Rightarrow x^2+x-2=0 \Rightarrow x=1$$$ and $$$x=-2$$$ where we have ensured that the only exponent that makes a power $$1$$ is the zero exponent, for any base.

To construct one of this type, it is enough to raise any basis to an equation and equal it to $$1$$. For example, choosing basis $$8$$ and equation $$$3x^2-9$$$ we obtain: $$$8^{3x^2-9}=1$$$ which will be solved as $$$3x^2-9=0 \Rightarrow \displaystyle x=\frac{\pm\sqrt{-4\cdot 3 \cdot (-9)}}{2\cdot 3}=\frac{\pm6\sqrt{3}}{6}=\pm \sqrt{3}$$$