Refresh your familiarity with the basic rules of logarithms:
log(xy) = log(x) + log(y)
log(x/y) = log(x) - log(y)
log(x^p) = p * log(x)
Apply the change of base formula, where necessary. Of most value is the conversion to the natural log: ln(x) = log(x) / log(e). In that equation, e is Euler's Number, 2.781828..., and the two log functions can be of any base, as long as they are the same base.
Use the definition of the derivative of the natural log to simplify differential calculus problems appropriately. The derivative of the natural log is given by d/dx (ln(x)) = 1/x, while the derivative of any arbitrary logarithm is given by d/dx (log_base_a(x)) = (1 / x) * (1 / ln(a)).
Use the logarithm for a substitution in integrals, where appropriate. For example, in integral calculus the integral of [(1/x) * (1/ln(x)) dx ] can benefit from the substitution u = ln(x), with du = (1/x) dx, so that the original integral becomes [(1/u) du].
Consider using logarithmic differentiation when differentiating a polynomial expression where the product rule would be complex. Taking the logarithm of each side and then differentiating can be an effective intermediate step.