Laws of Logarithms

Theorem

Let $x, y, b \in \R_+^*$ be strictly positive real numbers and let $a\in \R$ be any real number.

Then:

$(1): \quad \log_b x = \dfrac {\log_a x} {\log_a b}$

$(2): \quad \log_b x + \log_b y = \log_b \left({x y}\right)$

$(3): \quad \log_b \left(x^a\right) = a \log_b x$

$(4): \quad \log_b x - \log_b y = \log_b \left(\dfrac{x}{y}\right)$

where $\log_b$ denotes the logarithm to base $b$.

Proof

The proofs are somewhat different for general logarithms and natural logarithms.

General Logarithm

In the case of a general logarithm, which is what is typically examined in pre-university algebra and pre-calculus classes when these laws are first introduced, the proofs are based on the corresponding parts of the Laws of Exponents:

$(1): \quad$ See Change of Base of Logarithm

$(2): \quad$ See Sum of Logarithms

$(3): \quad$ See Logarithms of Powers

$(4): \quad$ Follows from $(2)$ and $(3)$ and the definition of division as multiplication by a reciprocal

$\blacksquare$

Natural Logarithm

In the more advanced case of a natural logarithm, the proofs of the second and fourth parts are substantially more complex because we cannot defer to the laws of exponents, as the laws of exponents are based on the laws of the natural logarithm, just as exponents themselves are formally defined in terms of the natural logarithm:

$(1): \quad$ Same as above: Change of Base of Logarithm

$(2): \quad$ See Sum of Logarithms: Proof for the Natural Logarithm

$(3): \quad$ See Logarithms of Powers: Proof for the Natural Logarithm

$(4): \quad$ Same principle as above

$\blacksquare$