Examples of exponential equations

Let us see the first two methods in action: reduction to the same base and substitution.

Example — Same base

Solve $2^{x+1} = 8^{x-1}$.

Since $8 = 2^3$, we have $8^{x-1} = 2^{3(x-1)} = 2^{3x-3}$. By injectivity of the exponential: $x + 1 = 3x - 3 \iff -2x = -4 \iff x = 2.$

Example — Substitution

Solve $4^x - 3\cdot 2^x + 2 = 0$.

Observe that $4^x = (2^2)^x = (2^x)^2$. Setting $t = 2^x$ with $t > 0$: $t^2 - 3t + 2 = 0 \iff (t-1)(t-2) = 0 \iff t = 1 \ \vee\ t = 2.$ Both solutions are positive, hence acceptable. Returning to $x$:

• $2^x = 1 = 2^0 \implies x = 0$;
• $2^x = 2 = 2^1 \implies x = 1$.

Links

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