Solving Exponential Equations

Recall that the function is one-to-one; therefore, if , we know b = c. Whenever possible, we’ll use this property to solve equations with indeterminate exponents.

Example 1

2x = 3x + 1

x = -1

If we can transform the bases so that they are identical, we can still use the principle above.

Example 2

-2x-1 = 2x-6

x =

When the bases are incompatible, we can use the Exponent Rule for Logarithms (preferably with natural logarithms or base-10 logarithms) and solve with a calculator.

Example 3

(2x - 11)ln3 = (4x - 5)ln7

(2ln3 - 4ln7)x = (11ln3 - 5ln7)

While this value for x is kind of unwieldy, it’s easy to obtain a decimal approximation with a scientific calculator.