The Graph of a Logarithmic Function

A logarithm answers the question an exponential asks in reverse: log base b of x is the power you raise b to in order to get x. Since b raised to any real power only ever produces a positive number, x must be positive for log base b of x to exist. That's the whole domain: x > 0. So as x creeps toward 0 from the right, the exponent needed to produce such a tiny positive number plunges toward negative infinity. The graph doesn't stop or turn around, it just keeps diving, hugging the y-axis forever without touching it. That vertical line x = 0 is the asymptote, and it exists purely because of the domain restriction, not because of any special algebra trick.

Once past that asymptote, the graph rises, crosses the x-axis at x = 1 (since log base b of 1 = 0 for any valid base), and keeps climbing as x grows. For b > 1, it bends downward, concave down, rising fast at first then flattening its rate of climb, though it never actually flattens into a horizontal line. That last point trips people up: the growth rate slows dramatically, but slow growth is not zero growth, so the function keeps inching upward without bound.

This graph is the mirror image of its inverse exponential function, reflected across the line y = x. Where the exponential has a horizontal asymptote and explodes upward, the log has a vertical asymptote and crawls upward. Keeping that inverse relationship in mind makes transformations of log graphs, shifts, stretches, reflections, much easier to predict.