Solution methods

To solve an exponential equation one always tries to reduce it to a form in which one can “remove the base” or substitute a new unknown.

In summary — Main methods

1. Same base. If both sides can be written as powers with the same base, for example $a^{f(x)} = a^{g(x)}$, then — by the injectivity of the exponential function — one can “remove the base” and reduce the equation to $f(x) = g(x)$.
2. Substitution. If an expression of the form $a^x$ appears several times in the equation (or $a^{2x} = (a^x)^2$), one sets $t = a^x$ (with the condition $t > 0$) and solves an equation in $t$ — often a quadratic. Finally one returns to $x$ by imposing $a^x = t$ for each root $t > 0$ found.
3. Passing to the logarithm. If nothing better can be done, one applies the logarithm (next chapter) to bring the exponent $x$ into “linear” form. We shall see this after having defined the logarithms.

The first two methods are the ones we shall use straight away; the third requires the logarithm and will be dealt with in the following chapter.

Links

Topics: Exponential function
Concepts: Injectivity
Functions: Exponential function
Methods: Exponential equations
Skills: Solving equations