Solving Equations Using Logarithms
Like the exponential functions, the logarithmic functions are one-to-one. Thus, if 
, then x = y. We can use this property to solve equations involving logarithms with the same base:
Example 1
When we use this property, it’s important to make sure the possible values for x are in the domain of the function. Note that -4 is not a solution to the original equation in example 1, even though it is a solution to 2x2 = 32.
Example 2
The final equation implies that x = 5 or x = -3 are possible solutions. However, x = -3 is not in the domain of 
; thus it is not a possible solution. The value x = 5 an element of the domain, and so is a solution to the equation.
Logarithms can also be useful when solving exponential equations. When we have an exponential equation including expressions with different bases, we might use a “computer-friendly” logarithm (like base e or base 10) to find the solution.
Example 3
This value for x looks complicated, but it can easily be approximated with a scientific calculator.