Exponents are just a shorthand for long multiplication problems. Instead of writing out x · x · x · x · x · x we would write x6, i.e. the variable and the number of times we wanted it to be multiplied by itself. The superscript part of the expression is called the exponent; the lower part, the x in our example, is called the base. Of course, we can do this with numbers as well as variables so, for example, 5 · 5 · 5 · 5 would be 54.
Logarithms are a little trickier to define using English. The logarithm of a number is the number to which you'd have to raise ten to get the original number. The multiple uses of "a number" make the definition a little hard to follow. It's usually a little easier to look at some examples.
log(1000) = log(103) = 3
log(100) = log(102) = 2
log(10) = log(101) = 1
log(1) = log(100) = 0
log(.1) = log(10-1) = -1
So how about numbers that aren't a power of 10? For those you usually need to get out your calculator. The thinking goes like this:
log(12) = 1.079 because 101.079 = 12
log(.12) = -0.9208 because 10-.09208 = .12
Why does the logarithm always represent a power of 10? It doesn't, necessarily. If no other base is specified then the assumption is that it's 10 but you can specify other bases by using a subscript. For example log2x is "the number to which you have to raise 2 to get x". Revisiting our previous examples
log2(8) = log2(23) = 3
log2(4) = log2(22) = 2
log2(2) = log2(21) = 1
log2(1) = log2(20) = 0
log2(.5) = log2(2-1) = -1
So what about log25.4? Or some other number that isn't a power of 2? Here you can run into problems since calculators don't have keys for every possible base. To calculate these values you can use a simple formula:
Applying this to my question at the beginning of the first paragraph, and using a calculator to get the base 10 logarithm values, gives us
|log25.4 =||log(5.4)||=||0.7323||= 2.432|
You can confirm with a scientific calculator that 22.423 = 5.4.