Logarithm is a mathematical way of expressing a number. Adding the logarithms of two numbers has the effect of multiplying the original numbers; subtracting them divides them, and many other useful things can be done too.
Algorithm is a set of instructions for doing something -- for example, a computer programme, street directions, or a recipe.
If ten to the power of X is Y, then X is the logarithm (to base 10) of Y. e.g. the logarithm of 100 is 2 the logarithm of 1000 is 3 the logarithm of 316 is 2.5