How do I prove using the definition that the logarithmic function is continuous

Hint: Since the logarithm function satisfies $$ f(xy) = f(x)+f(y) $$ for any $x,y\in\mathbb{R}^+$, in order to prove the continuity over $\mathbb{R}^+$ you just need to prove the continuity in $1$, since: $$ \log(x+\varepsilon)-\log(x) = \log\left(1+\frac{\varepsilon}{x}\right).$$ Now the continuity in $1$ follows from the Bernoulli inequality: $$ \forall x\in(-1,1),\quad x+1\leq e^x \leq \frac{x}{x-1} \tag{1}$$ (proving $(1)$ does not necessarily depends on the continuity of the exponential function. For instance, we can prove $(1)$ for any $x\in(-1,1)\cap\mathbb{Q}$ by induction) and a straightfoward consequence of $(1)$ is: $$ \frac{y}{1+y}\leq \log(1+y) \leq y \tag{2} $$ for any $y$ in a neighbourhood of zero. Continuity follows.