A Different Perspective

Basic logarithmic functions, are those functions which CAN be represented by formulas of the form \(A \, \log _r(B \, x + C) + D\).

We can decide whether the function is increasing or decreasing by the value of \(r\), the sign of \(A\), and the sign of \(B\).

• \(r > 1\) and \(A > 0\) and \(B > 0\) : increasing positive function
• \(r > 1\) and \(A > 0\) and \(B < 0\) : decreasing negative function
• \(r > 1\) and \(A < 0\) and \(B > 0\) : decreasing positive function
• \(r > 1\) and \(A < 0\) and \(B < 0\) : increasing positive function
• \(r < 1\) and \(A > 0\) and \(B > 0\) : decreasing positive function
• \(r < 1\) and \(A > 0\) and \(B < 0\) : increasing negative function
• \(r < 1\) and \(A < 0\) and \(B > 0\) : increasing positive function
• \(r < 1\) and \(A < 0\) and \(B < 0\) : decreasing positive function

On the other hand, we have the algebra rule \(\frac {1}{b} = b^{-1}\). We could think of the base of the logarithmic formula as always being greater than \(1\), and just use positive and negative exponents to switch between increasing and decreasing functions.

We can rewrite and logarithmic expression with an equivalent expression that has a base greater than \(1\).

Suppose \(0 < r < 1\).

Then \(\frac {1}{r} > 1\). We would like to use this new base, which we can accomplish through the change of base formula.

\(\log _{\tfrac {1}{r}}(r)\) is the thing you raise \(\frac {1}{r}\) to, to get \(r\), which is \(-1\).

That gives us

\(\blacktriangleright \) We can switch and logarithm from a base less than \(1\), to a base greater than \(1\) just by negating the logarithm.

New Idea: Basic logarithmic functions, are those functions which CAN be represented by formulas of the form \(A \, \log _r(B \, x + C) + D\), where \(r > 1\).

In this case, we would have the following behaviors:

• \(A > 0\) and \(B > 0\) : increasing positive function
• \(A > 0\) and \(B < 0\) : decreasing negative function
• \(A < 0\) and \(B > 0\) : decreasing positive function
• \(A < 0\) and \(B < 0\) : increasing positive function

We would decide function behavior (increasing or decreasing) by the signs of both leading coefficients, \(A\) and \(A\).

\(\blacktriangleright \) If \(A\) and \(B\) are the same sign, then we have an increasing function.

\(\blacktriangleright \) If \(A\) and \(B\) are different signs, then we have a decreasing function.

(e)

In this model, we are using bases that are greater than \(1\).

If this is the case, then we might as well use \(e\) as our base.

\(\blacktriangleright \) Basic logarithmic functions, are those functions which CAN be represented by formulas of the form \(A \, \ln (B \, x + C) + D\).

In this model, our basic forms to memorize would be

Pick Your Form

You should pick your own logarithmic exponential form that you like and understand.

Then, you can change anything given to you into that form.

You might pick a particular number greater than \(1\) as the base you like. \(e\) is a very popular choice, because it shows up quite frequently in mathematics, like Calculus.

If you pick a base you like and change formulas to use that base, then you have a better chance of quickly deciding on function characteristics.

Basic Basic Form: \(\ln (x)\)

Basic Forms: \(\ln (x)\), \(\ln (-x)\), \(-\ln (x)\), \(-\ln (-x)\)

Now increasing and decreasing become questions about the signs of the two leading coefficents - the leading coefficient of the entire formula and the leading coefficient of the exponent.