Logarithmic Functions

Press a button on your phone and you raise a number to a power. Logarithms are the undo button. They run the whole thing backwards and ask the only question that matters: “what power was used to get here?” That is the entire idea. A logarithm is a power detective. Once you see it that way, every graph, asymptote and equation in this topic becomes a story about powers that have been hidden and need finding.

What a logarithm actually asks

Start with a power statement you already trust, like $e^3 = 20.08\ldots$. Reading it forwards you say ”$e$ to the power $3$”. Reading it backwards a logarithm says “the power that turns $e$ into $20.08\ldots$ is $3$”, which we write as $\log_e(20.08\ldots) = 3$. The base of the log is the number being raised, and the answer of the log is the power.

So $\log_a(x)$ is the power you raise $a$ to in order to get $x$. Two facts fall straight out of that sentence and they fix most mistakes before they start. First, you can only ask for the power that produces a positive number, because $a^{\text{anything}}$ is always positive. That is why the input of every log must be greater than zero. Second, a log and its matching power statement are two ways of saying the same thing:

The star of Methods is the natural logarithm, written $\log_e(x)$, whose base is the special number $e \approx 2.718$. When you see a log with no base shown in this course, treat it as base $e$.

The shape of the graph and its wall

Picture the graph of $y = \log_e(x)$. It climbs forever to the right, but slower and slower, like a runner who never quite stops. It crosses the $x$ axis at the point $(1, 0)$, because $e^0 = 1$ means the power that makes $1$ is zero.

The interesting bit is on the left. As $x$ shrinks towards zero the graph plunges downward without limit, hugging the $y$ axis but never touching it. That invisible wall is the vertical asymptote, here the line $x = 0$. It appears because the input of a log can never reach zero, so the curve has nowhere to land. Every basic log graph has exactly one vertical asymptote, and finding where it sits is worth easy marks.

Logs and exponentials are mirror twins

Here is the cleanest way to understand logs. The function $y = \log_e(x)$ is the inverse of $y = e^x$. They undo each other, so $\log_e(e^x) = x$ and $e^{\log_e(x)} = x$. On a graph, inverse functions are reflections of each other in the line $y = x$, which is why the log graph is just the exponential graph flipped across that diagonal.

That reflection swaps everything. The exponential has a horizontal asymptote at $y = 0$, so the log has a vertical asymptote at $x = 0$. The exponential takes any $x$ and outputs only positives, so the log takes only positives and outputs any $y$. Domain and range trade places. To find the rule of a log inverse, swap $x$ and $y$ in the exponential equation, then make $y$ the subject by taking $\log_e$ of both sides.

Moving and stretching the graph

Transformations of $y = \log_e(x)$ follow the same rules as any function, but two features need watching because examiners target them every year.

1. A change inside the bracket shifts the graph left or right and drags the vertical asymptote with it. For $y = \log_e(x - h)$ the asymptote moves from $x = 0$ to $x = h$. The sign flips your instinct, so $\log_e(x - 4)$ moves right and the asymptote lands at $x = 4$.
2. A change outside the log, added or multiplied, shifts or stretches the graph up and down. This never moves a vertical asymptote, but students forget the vertical translation constantly, so write it down.

To solve a log equation, get a single log by itself, rewrite it as a power of the base to peel the log away, then solve what is left. Finish by checking every answer against the domain. Reject any value that makes an argument zero or negative, because it was never a real solution, only a ghost created by the algebra.

See these moves in action in the Worked Examples tab, then test yourself in Try It.

Lock it in with active recall