Logarithm on Positive Real Numbers is Group Isomorphism

Theorem

Let $\struct {\R_{>0}, \times}$ be the multiplicative group of positive real numbers.

Let $\struct {\R, +}$ be the additive group of real numbers.

Let $b$ be any real number such that $b > 1$.

Let $\log_b: \struct {\R_{>0}, \times} \to \struct {\R, +}$ be the mapping:

$x \mapsto \map {\log_b} x$

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

Then $\log_b$ is a group isomorphism.

Proof

From Sum of Logarithms we have:

$\forall x, y \in \R_{>0}: \map {\log_b} {x y} = \map {\log_b} x + \map {\log_b} y$

That is $\log_b$ is a group homomorphism.

From Change of Base of Logarithm, $\log_b$ is a constant multiplied by the natural logarithm function.

Then we have that Logarithm is Strictly Increasing.

From Strictly Monotone Real Function is Bijective, it follows that $\log_b$ is a bijection.

So $\log_b$ is a bijective group homomorphism, and so a group isomorphism.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 47$. Homomorphisms and their elementary properties: Illustration