The logarithmic function

A logarithmic function is $\displaystyle {f(x)}={\log_a x}$, where $\displaystyle a>0$ and $\displaystyle a\neq1$. A table of values of $\displaystyle {f(x)}={\log_2 x}$:

• When $\displaystyle x$ doubles, the value goes up by 1: the graph rises, but slowly.
• When $\displaystyle x$ comes near 0, the value goes down without limit.

The drawing shows the graphs of $\displaystyle {f(x)}={\log_2 x}$ and of $\displaystyle {g(x)}={\log_{\frac{1}{2}}x}$. Each is the reflection of the other in the x-axis: $\displaystyle {\log_{\frac{1}{2}}x}={-\log_2 x}$.

• Domain: $\displaystyle x>0$. Neither 0 nor a negative number has a logarithm, so the whole graph is to the right of the y-axis.
• The point on the x-axis: $\displaystyle (1,0)$, since $\displaystyle {\log_a 1}={0}$. No point on the y-axis.
• Vertical asymptote: $\displaystyle {x}={0}$ – the y-axis. The graph comes near it as $\displaystyle x$ comes near 0.
• $\displaystyle a>1$: the function increases throughout its domain; positive where $\displaystyle x>1$, negative where $\displaystyle 0<x<1$.
• $\displaystyle 0<a<1$: the function decreases throughout its domain; positive where $\displaystyle 0<x<1$, negative where $\displaystyle x>1$.

$\displaystyle \log_2 x$ undoes $\displaystyle 2^x$: the point $\displaystyle (1,2)$ is on the graph of $\displaystyle 2^x$, and the point $\displaystyle (2,1)$ on the graph of $\displaystyle \log_2 x$.

• Every point $\displaystyle (p,q)$ on the graph of $\displaystyle a^x$ matches the point $\displaystyle (q,p)$ on the graph of $\displaystyle \log_a x$.
• So the two graphs are reflections of each other in the line $\displaystyle {y}={x}$.
• Their properties swap: the horizontal asymptote $\displaystyle {y}={0}$ of $\displaystyle a^x$ is the vertical asymptote $\displaystyle {x}={0}$ of $\displaystyle \log_a x$; the point $\displaystyle (0,1)$ is the point $\displaystyle (1,0)$.

The drawing shows four functions, from $\displaystyle \log x$ to $\displaystyle 3\log x+4$:

• $\displaystyle {g(x)}={\log(x-1)}$: a horizontal shift of 1 to the right. The domain moves too – $\displaystyle x>1$ – and so does the asymptote: $\displaystyle {x}={1}$. The point on the x-axis is $\displaystyle (2,0)$.
• $\displaystyle {h(x)}={3\log x}$: a vertical stretch by 3. The domain, the asymptote $\displaystyle {x}={0}$ and the point $\displaystyle (1,0)$ stay.
• $\displaystyle {k(x)}={3\log x+4}$: a vertical shift of 4 up. The point on the x-axis: $\displaystyle {3\log x+4}={0}$, that is $\displaystyle {\log x}={-\frac{4}{3}}$, and $\displaystyle {x}={10^{-\frac{4}{3}}\approx0.046}$.

The drawing shows $\displaystyle {f(x)}={\log_2 x}$ and three operations on it:

• $\displaystyle {g(x)}={\log_2(-x)}$: a reflection in the y-axis. The domain: $\displaystyle -x>0$, that is $\displaystyle x<0$.
• $\displaystyle {h(x)}={-\log_2 x}$: a reflection in the x-axis. The function decreases.
• $\displaystyle {k(x)}={|\log_2 x|}$: the part below the x-axis (where $\displaystyle 0<x<1$) goes up above it. $\displaystyle k(x)\geq0$, and its minimum is at the point $\displaystyle (1,0)$.

$\displaystyle \log_a f(x)$ is defined only where $\displaystyle f(x)>0$. The example below finds the domain of $\displaystyle \log(x^2-4)$:

• The parabola $\displaystyle x^2-4$ is positive outside its roots $\displaystyle -2$ and $\displaystyle 2$: $\displaystyle x<-2$ or $\displaystyle x>2$.
• As $\displaystyle x$ comes near $\displaystyle 2$ or $\displaystyle -2$ from inside the domain, $\displaystyle x^2-4$ comes near 0, and the logarithm goes down without limit: the graph has two vertical asymptotes, $\displaystyle {x}={2}$ and $\displaystyle {x}={-2}$.

The function is $\displaystyle {f(x)}={\log_2(x+2)}$.

Find the domain and the vertical asymptote.

Mark the points where the graph meets the axes.

The graph of $\displaystyle {y}={\log_a x}$ passes through the point $\displaystyle (9,2)$.
Write the function.

The function is $\displaystyle {f(x)}={\log_6(x^2-5x)}$.

Find the domain of the function.

Write the equations of the vertical asymptotes.

For which values of $\displaystyle x$ is $\displaystyle {f(x)}={1}$?

Mark the points where the graph of $\displaystyle f$ meets the line $\displaystyle {y}={1}$.

Compare $\displaystyle {f(x)}={\log_3 x}$ with $\displaystyle {g(x)}={\log_{\frac{1}{3}}x}$: the domain, the points on the axes, where each is positive and negative, where each increases and decreases, the asymptotes.