Laws of Logarithms

We've remarked that log notation can be used to express any exponential relationship  y = f(x) = b^x  in the inverse – but equivalent – form:  x = g(y) = log_b y .

This equivalent form of expression lets us put many familiar properties of exponentials in logarithmic notation.

For example,  b⁰ = 1  becomes  0 = log_b 1 . Likewise,  b¹ = b  becomes  1 = log_b b .

Other familiar laws of exponents can be re-expressed as laws of logarithms:

bmbn = bm + n	becomes	logb (mn) = logb m + logb n .
bm / bn = bm – n	becomes	logb (m / n) = logb m – logb n .
(bm)n = bmn	becomes	logb (mn) = n logb m .

If these don't seem quite obvious, consider the first law above. Let  x = b^m  and  y = bⁿ . Then  m = log_b x  and  n = log_b y . The exponential form of the law says that  xy = b^mbⁿ = b^{m + n} . In other words,  log_b (xy) = m + n . If we put this together with  m = log_b x  and  n = log_b y  we get  log_b (xy) = log_b x + log_b y . This is the logarithmic form of the law.

There's nothing mysterious going on here. It's just a matter of notation.

Can you use the last two laws of exponents to arrive at the last two laws of logarithms in a similar fashion?