Topic 2.11 Notes – Logarithmic Functions

What Logarithmic Functions Are

A logarithmic function in general form is

$$f(x)=a\log_b x$$

with $a\ne 0$, $b>0$, and $b\ne 1$.

The core meaning is

$$\log_b x=y \iff b^y=x$$

So a logarithm asks, “What exponent on base $b$ gives $x$?”

Because logs and exponentials are inverses, their graphs reflect across $y=x$, as in the graph of $2^x$ and $\log_2 x$ below.

Useful anchor points for $y=\log_b x$:

• $(1,0)$ because $\log_b 1=0$
• $(b,1)$
• $\left(\frac1b,-1\right)$

For $f(x)=a\log_b x$, those become:

• $(1,0)$
• $(b,a)$
• $\left(\frac1b,-a\right)$

A couple names you should recognize:

• Common log means $\log x=\log_{10}x$
• Natural log means $\ln x=\log_e x$

In general form, the domain is $(0,\infty)$ and the range is $(-\infty,\infty)$.

Intercepts:

• $x$-intercept is always $(1,0)$
• No $y$-intercept, because $x=0$ is not in the domain

If the argument is changed, like $\log_b(kx+c)$, the domain comes from the full argument being positive:

$$kx+c>0$$

How the Graph Behaves

Logs are monotonic on their whole domain. They never switch from increasing to decreasing.

Increasing and decreasing

For $y=\log_b x$:

• if $b>1$, it is increasing
• if $0<b<1$, it is decreasing

If $a<0$, multiplying by $a$ reflects across the $x$-axis, so the direction reverses.

Compact test for $f(x)=a\log_b x$:

• increasing when $\frac{a}{\ln b}>0$
• decreasing when $\frac{a}{\ln b}<0$

Concavity

For the parent graph:

• if $b>1$, it is concave down
• if $0<b<1$, it is concave up

If $a<0$, concavity reverses too.

That gives an important general-form pairing:

• increasing logs are concave down
• decreasing logs are concave up

Extrema and inflection points

On the full domain, logarithmic functions have:

• no local or absolute extrema
• no inflection points

On a closed interval, any max or min happens at an endpoint.

Asymptotes, End Behavior, and Transformed Arguments

In general form, the vertical asymptote is

$$x=0$$

You only approach it from the right, so use $x\to 0^+$, not a two-sided limit.

For an increasing log:

$$\lim_{x\to 0^+}f(x)=-\infty \qquad \lim_{x\to\infty}f(x)=\infty$$

For a decreasing log:

$$\lim_{x\to 0^+}f(x)=\infty \qquad \lim_{x\to\infty}f(x)=-\infty$$

There is no horizontal asymptote because outputs stay unbounded.

For

$$g(x)=a\log_b(kx+c)+d$$

here is what changes:

• domain comes from $kx+c>0$
• vertical asymptote is where $kx+c=0$, so

$$x=-\frac{c}{k}$$

• the graph exists only on the side where $kx+c>0$
• $d$ shifts up or down, but does not change range, monotonicity, or concavity
• if $k<0$, increasing/decreasing reverses, but concavity does not

So transformed logs can be any of these:

• increasing, concave down
• decreasing, concave down
• increasing, concave up
• decreasing, concave up

Example:

$$g(x)=\log_2(6-2x)$$

Domain:

$$6-2x>0 \Rightarrow x<3$$

So the asymptote is $x=3$, and the graph is on the left of it. Since $6-2x$ decreases as $x$ increases, the graph is decreasing. It stays concave down. The graph below also shows the x-intercept at $\left(\frac52,0\right)$.

Recognizing Logarithmic Structure from Tables, Equations, and Graphs

In an equation, the variable is inside a logarithm, and the full argument must stay positive.

In a graph, look for these clues:

• one side of a vertical asymptote only
• monotonic on the whole domain
• one concavity only
• unbounded near the asymptote and also at the far end
• no extrema and no inflection points

In a table, the hallmark is this: equal output changes match proportional inputs.

Example for $f(x)=2\log_3 x$:

• outputs $-2,0,2,4$
• inputs $\frac13,1,3,9$

Each output goes up by 2, and each input gets multiplied by 3.

With shifted logs, raw $x$-values often are not proportional. The proportional pattern may appear only after adjusting for the asymptote, like distances from it.

Common Mistakes on the Test

• Saying the domain is all real numbers for general form
• Giving a $y$-intercept at $x=0$
• Mixing up “undefined at $0$” with the one-sided limit as $x\to 0^+$
• Using a two-sided limit at $0$
• Assuming every log is increasing
• Keeping the general-form increase/concavity pairing after a horizontal reflection inside the argument
• Checking raw $x$-values for proportionality in shifted logs instead of distances from the vertical asymptote