Powers with the same base are equal, if and only if their exponents are equal as well.
See solution.
Practice makes perfect
We can solve an exponential equation with unlike bases by rewriting it using the same base or by graphing. We will explain both approaches individually.
Rewriting the Equation Using the Same Base
Sometimes we can use the Properties of Exponents to rewrite the original equations such that both sides are powers with a common base. Then, according to Property of Equality for Exponential Equations, we can set the exponents equal to each other and solve.
For example, consider the equation shown below.
1/2^x = 4
We can use the definition of a negative exponent to rewrite the left-hand side as 2^(- x), and since 4= 2^2 we can write both sides as a power with a common base of 2.
1/2^x = 4 ⇔ 2^(- x) = 2^2
Now, since both powers are equal and have the same base, their exponents must be equal as well. This allows us to solve for x.
2^(- x) = 2^2 ⇔ - x = 2
From the equation above, we can conclude that the solution for our original equation is x =- 2.
Solving by Graphing
Sometimes it is not possible to rewrite the exponential equation using a common base. In these cases we can still solve the equation by graphing. To do this, we need to graph both sides of the equation together. The x-coordinate of the intersection point is the solution to the equation. Consider the example shown below.
3(2)^x=3^x
First, note that it is not possible to rewrite this equation using two powers with a common base. Let's graph the exponential functions from both sides together.
From the graph above we can approximate the solution, x ≈ 2.7.
