Logarithms, written in the form are only defined for To prove this, it is necessary to go back to the definition of logarithms. The logarithm of a number is the exponent to which the logarithm's base must be raised in order to get the desired number. For instance, is the same as asking the question, "What is the exponent needed to raise to in order to get ?" Here, Suppose instead that the value of wanted to be found. Exponentially, this can be though of as The graph of can be used to visualize this equation.
From the diagram, it can be seen that never lies below the -axis. This means the function value is never negative. In fact, the function never equals — it gets closer and closer to To examine this, let's consider positive and negative values of
is multiplied by itself times. For example Since is a positive number, no matter how many times it is multiplied, the product will always be positive. Therefore, there are no positive -values that make negative.
Any base raised to a negative exponent can be rewritten as a fraction. Can be negative if is negative? A base with a negative exponent can be rewritten as a fraction.
The "more negative" becomes the smaller will be. It gets closer and closer to without actually equaling or becoming negative.
Therefore, there are no values of that make negative. Since has no real solution, is undefined.