A Notation for Teaching Exponentiation and Logarithms

Learning exponentiation and logarithms is inherently more difficult than learning addition, subtraction, multiplication and division. To make matters worse, the notation used for exponentiation and logarithms makes these operations even more difficult.

I am proposing a simple way of writing the exponentiation and logarithm operations that is consistent with the other operators. I am not suggesting that we discard the standard notation, but I think that learning can be made easier if the new notation is presented before introducing the standard notation. In this section I will define the new notation and present examples using both notations side by side in an effort to convince someone that the new notation is sufficiently more clear as to merit experimental study.

Exponential Notation

Getting Rid of the Superscript

There are two things wrong with the standard notation , x^y, used for exponentiation. I don't think that there will be much argument with my first criticsim, which is the use of a superscript in place of an operational sign. To get around this, x^y is sometimes written as x ^ y.

Order Reversal

You may initially disagree with my second criticism, which is that the base and exponent are presented in the wrong order. x^y should be written as y ^ x.

One reason for reversing the order is that it is consistent with the way that multiplication is treated. x + x = 2 * x. Therefore x * x should be 2 ^ x.

Logarithm Notation

There is a more compelling reason for reversing the order of exponential notation, which is that it allows for a convenient notation for logarithms.

In standard notation we have the following cumbersome notation:
The inverse operation of y = a^x is x = log_ay or log_a(a^x) = x.

Now since upward arrows are used for exponentiation, the most natural choice for an inverse operation would be to use a downward arrow. We therefore get:

The inverse of y = x ^ a is x = y ¯ a or (x ^ a) ¯ a = x, which is analogous to:
The inverse of y = x + a is x = y - a or (x + a ) - aa = x.

This would be even clearer if an upward dagger was used in place of "^". Unfortunately, Internet Explorer does not recognize the upward dagger symbol. This symbol: "­" will show up for Netscape but not Internet Explorer.

Side By Side Comparison

Two problems will be presented for comparison. Both problems use the following rule:
log(a^x) = x * (log a), or in the proposed notation, (x ^ a) ¯ 10 = x * (a ¯ 10).

Consider the following typical problem: Use logarithms to detemine the amount of money in the bank of a person who deposits $1 at 5% interest for 20 years.

In standard notation we have:
x = 1.05²⁰.
log(x)=log(1.05²⁰) = 20*log(1.05).
x = 10^20*log(1.05).

In the proposed notation we have:
x = 20 ^1.05.
x = (x ¯ 10) ^ 10 = ((20 ^1.05) ¯ 10) ^ 10 = (20 * (1.05 ¯ 10)) ^ 10.

Now determine how many years it will take this person to have $100.

In the standard notation we have:
(1.05)^x = 100.
log(1.05)^x = log(100)
x * log(1.05) = log(100).
x = log(100) / log(1.05).

In the proposed notation:
x ^ 1.05 = 100.
(x ^ 1.05) ¯ 10 = 100 ¯ 10.
x*(1.05 ¯ 10 ) = 100 ¯ 10.
x = (100 ¯ 10) / (1.05 ¯ 10 ).