Core Logarithmic Functions

The expression \(\log _a(b)\) was defined to be the number that you raise \(a\) to, to get \(b\).

We made a Core logarithmic function from this by fixing the base: \(L(x) = \log _r(x)\).

This function is the “reverse” of \(r^x\). Meaning that if \((A, B)\) is a pair in the \(L(x) = \log _r(x)\) function, then \((B, A)\) is a pair in the \(r^x\) function. Their domains and ranges are swapped.

• The domain of an exponential function is all real numbers. The range of a logarithmic function is all real numbers.
• The range of an exponential function is all positive real number. The domain of a logarithmic function is all positive real numbers.
• The graphs of exponential functions have a horizontal asymptote. The graphs of logarithmic functions have a vertical asymptote.

Here is the graph of \(y = L(x) = \log _2(x)\).

Here is the graph of \(y = E(x) = 2^x\).

The graphs of exponential functions (and shifted exponential functions) have a horizontal asymptote.

The graphs of logarithmic functions have a vertical asymptote.

We get general logarithmic functions by composing with linear functions.

General Logarithmic Functions

Logarithmic functions are functions taht can bre represented by formulas of the form

These templates can be viewed as a composition.

• Let \(CL(t) = \log _r(t)\) be a Core logarithmic function.
• Let \(L_{in}(u) = B \, u + C\) be a linear function.
• Let \(L_{out}(v) = A \, v\) be a linear function.

Then, our general logarithmic function can be expressed as a composition.

Now we can use the Chain Rule to establish whether a logarithmic function increases or decreases.

Logarithmic Function

Here is the graph of \(y = L(x) = -3 \log _2(5-x)+1\).

Analysis:

Domain: The inside must be positive. This tells us that the domain is \((-\infty , 5)\).

Zero:

\begin{align*} -3 \log _2(5-x)+1 &= 0 \\ \log _2(5-x) &= \frac {1}{3} \\ 5-x &= 2^{\tfrac {1}{3}} \\ 5-2^{\tfrac {1}{3}} &= x \end{align*}

Just Checking: \(5-2^{\tfrac {1}{3}} \approx 3.74 \in (-\infty , 5)\) and agrees with the graph.

Continuity: Logarithmic functions are continuous.

Behavior:

We can view \(L(x) = -3 \log _2(5-x)+1\) as a composition of three functions.

• Let \(CL(t) = \log _2(t)\), is an increasing Core logarithmic function.
• Let \(L_{in}(u) = 5 - u\), is a decreasing linear function.
• Let \(L_{out}(v) = -3 \, v + 1\), is a decreasing linear function.

Then, our general logarithmic function can be expressed as a composition.

\[ L(x) = -3 \log _2(5-x)+1 = (L_{out} \circ CL \circ L_{in})(x) \]

The Chain Rule tells us that \(dec \circ inc \circ dec = inc\)

\(L(x)\) is an increasing function.

Just Checking: This agrees with the graph.

End-Beavior: \(L\) only has end-behavior in only one direction.

The domain of \(L\) is \((-\infty , 5)\) and \(L\) is an increasing logarithmic function. The gives

\[ \lim \limits _{x \to -\infty } L(x) = -\infty \]

Singularity Beavior:

The domain of \(L\) is \((-\infty , 5)\) and \(L\) is an increasing logarithmic function. The gives

\[ \lim \limits _{x \to 5^-} L(x) = \infty \]

Global Maximum and Minimum: Logarithmic functions do not have a global maximum or global minimum.

Local Maximums and Minimums: Logarithmic functions do not have local maximums or minimums.

Range: The range of any logarithmic function is either \((-\infty , \infty )\).

Reading Coefficients

Behavior

Logarithmic functions either increase or they decrease.

The domain and behavior can tell you which.

This can also be summed up by comparing the base, the leading coefficient, and the leading coefficient of the linear inside function.

Our general template for logarithmic functions looks like

If the base \(r > 1\), then

• \(A > 0\) and \(B > 0\) gives an increasing logarithmic function.
• \(A < 0\) and \(B > 0\) gives a decreasing logarithmic function.
• \(A > 0\) and \(B < 0\) gives a decreasing logarithmic function.
• \(A < 0\) and \(B < 0\) gives an increasing logarithmic function.

If the base \(r < 1\), then everything is reversed (per the Chain Rule)

• \(A > 0\) and \(B > 0\) gives an decreasing logarithmic function.
• \(A < 0\) and \(B > 0\) gives a increasing logarithmic function.
• \(A > 0\) and \(B < 0\) gives a increasing logarithmic function.
• \(A < 0\) and \(B < 0\) gives an decreasing logarithmic function.

\(\blacktriangleright \) When the leading coefficients are the same sign, then the behavior of the logarithmic function is the same as the Core function.

\(\blacktriangleright \) When the leading coefficients are different signs, then the behavior of the logarithmic function is opposite the Core function.

Mental Models

Just like with exponential functions, it is nice to have a mental model in your head of a basic logarithmic function an its characteristics.

And, like with exponential functions, we like \(e\) as the base.

This gives us a basic basic logarithmic function: \(\ln (x)\)

And, then three other alternative choices, if you prefer.