Logarithmic Functions

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

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

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 \(y = L(x) = \log _2(x)\).

The graph of a basic logarithmic function has a vertical asymptote where the inside of the logarithm equals \(0\). The domain includes only numbers that make the inside of the formula positive. The function increases over its domain and is unbounded.

Shifted Logarithmic Function

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

The inside of the logarithm here is \(5-x\) and this equals \(0\) when \(x=5\). Therefore, the vertical asymptote in the graph is \(x=5\). The domain is \((-\infty , 5)\), since these are the numbers that make the inside of the logarithm formula positive.

Behavior

Our general template for logarithmic functions looks like

Of we choose \(e\) as the base, then they look like

• \(A\) is the leading coefficent for the function.
• \(B\) is the leading coefficent of the inside (the argument).

Comparing these back to our basic logarithm functions, we get

• \(A > 0\) 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.

\(\blacktriangleright \) When the leading coefficients are the same sign, then the logarithmic function is increasing.

\(\blacktriangleright \) When the leading coefficients are different signs, then the logarithmic function is decreasing.

Basic 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.