The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 10 10 = 103. More generally, if x = by, then y is the logarithm of x to base b, and is written y = logb(x), so log10(1000) = 3 .
The logarithm of a product is the sum of the logarithms of the factors:
Similarly, the logarithm of the ratio of two numbers is the difference of the logarithms.
To prove this, let m = logxa.
Rewrite the above expression as an exponent. xm = a (logxa asks " x to what power is a ?" And the equation answers: "x to the m is a.")
Let n = logxb. Thus, xn=b.
If we replace a and b based on the previous equations, we get:
This can be further simplified to:
Which, using the first law of exponents, can be written as:
This is the key step. Therefore, it can be seen that the properties of logarithms come directly from the laws of exponents. Replacing m and n with what they were originally defined as results in this equation:
Hence, the previous problem has been proven.