# Extension E: What are logs and exponents?

Consider the simple function b The log is the inverse function of the exponent. It can be applied to the value X to determine the exponent (n) at a given base. So to find the exponent (n) that raises b to give a specific value of X, we take the Log of X. Thus log Logs and exponents are therefore
As log(x) increases by 1 the value of x increases by multiples of 10. So an increase of 1 in the log increases x by a factor of 10, an increase of 2 in the log increases X by a factor of 100 (10 * 10), an increase of 3 in the log increase x by a factor of 1000 (10 * 10 * 10) and so on. The key fact to extract here is that increasing X by The natural log is the one where the base is approximately 2.718. This base has
mathematical properties that make it useful in a variety of situations relating to calculus. Logarithms can be defined to any positive base other than 1, not just x) or sometimes, if the base e is implicit (as it is here), simply log(x). An important fact about logs, in terms of logistic regression, is that because they represent powers of a base value (as we see in the above table) this allows us to translate multiplication into addition of logarithms. Two properties follow from this, namely:
This means the logistic regression equation can be linear and additive for the logged odds:
but multiplicative for the odds:
This is why the regression coefficients (b) can be interpreted in terms of odds ratios, by taking the exponential of the log odds [Exp(b)].
An explanation of logistic regression begins with an explanation of the logistic function:
The input is z= where a is the intercept and b1, b2, b3 to b |