For logarithms, as a supplement to the site that @PhiNotPi found, it may help to think of logarithms this way:

In the beginning there was addition. You could add two numbers like 7 and 5 to get 7 + 5. Then people realized you could add more than two numbers, like 7 + 5 + 3. It was found that in some cases you added the same number multiple times, like 3 + 3 + 3 + 3 + 3. Thus multiplication was born, designating this as 3×5. As with addition, the same number could be multiplied multiple times, like 3×3 x 3×3 x 3. This is called exponentiation, 3 ^ 5.

Exponentiation is different in some ways from addition and multiplication. Whereas 3 + 2 = 2 + 3 and 3×2 = 2×3, 3^2 = 9 and 2^3=8. In general, the order of the terms makes a difference. We need names to distinguish them. The first is called the base and the second is called the exponent or logarithm. If we want to solve a problem like 5^x = 25, we describe that as finding the log of 25 base 5. In this case x = 2, because 5^2 = 5×5 = 25.

The rules for combining multiplication and logarithms are analogous to the rules for combining addition and multiplication. For example, we have the distributive law for addition and multiplication, (5×3) + (2×3) = (3+3+3+3+3) + (3+3) = 7×3. In general, a(b+c) = a x b + a x c.

In the case of logarithms, we have (7^3) x (7^2) = (7×7x7) x (7×7) = 7^5. In general (a^b) x (a^c) = a^(b+c). We can also go in the reverse direction. If we say u=7^3 and v = 7^2, then we have log(u x v) = log(u) + log(v), where we require that we are always working with the same base, 7 in this case.

This last property was the motivation for the invention of logarithms by John Napier in the beginning of the 17th century. If you have a table of logarithms, to base 10 for convenience, you can use it to multiply two numbers by a combination of table lookup and addition. To multiply two numbers, a and b, look up the logarithms of a and b in the table. Then add to get log(a) + log(b). To find c such that c = a x b, look in the table to find the number c such that log( c) = log(a) + log(b). The log table only needs to have entries for numbers from 1 to 10, since, for example, 123 = 1.23×10 x 10, so log(base 10) 123 = 2 + log(base 10)1.23.