Appendix 4

Exponential Functions of Different Bases

To see how exponents of one base can yield exponents of other bases, we need to make use of an important property of exponents that is often called the power rule. To see how it works, let’s look at a few examples of exponents raised to an additional power.

(23)2: This means to raise 23 to the second power, so we have a power raised to another power.

Doing so yields (23)2 = (23)(23).

This equals (2 × 2 × 2)(2 × 2 × 2) because 23 means to multiply three 2s together.

Thus, (23)2 = (2 × 2 × 2)(2 × 2 × 2).

Dropping parentheses gives

image

This means that (23)2 is the equivalent of multiplying six 2s together.

Thus, (23)2 equals 26 or (23)2 = 23×2. (This is the power rule.)

(54)3: This means to raise 54 to the third power.

Doing so yields (54)3 = (54)(54)(54).

This equals (5 × 5 × 5 × 5)(5 × 5 × 5 × 5)(5 × 5 × 5 × 5) because 54 means to multiply four 5s together.

Thus, (54)3 = (5 × 5 × 5 × 5)(5 × 5 × 5 × 5)(5 × 5 × 5 × 5).

Dropping parentheses gives

image

This means that (54)3 is the equivalent of multiplying 12 5s together.

Thus, (54)3 equals 512 or (54)3 = 54×3. (This is the power rule.)

(22)x: Using the power rule, we have (22)x = 22 · x = 22x.

But we also know that (22)x = (4)x = 4x (because 22 equals 4).

Thus, 22x = 4x.

Note that 22x has a base of 2 and exponent of 2x.

This means that 4x can be written as an exponent that has a base of 2.

(23)x: Using the power rule, we have (23)x = 23 · x = 23x.

But we also know that (23)x = (8)x = 8x (because 23 equals 8).

Thus, 23x = 8x.

Note that 23x has a base of 2 and exponent of 3x.

This means that 8x can be written as an exponent that has a base of 2.

(25)x: Using the power rule, we have (25)x = 25 · x = 25x.

But we also know that (25)x = (32)x = 32x (because 25 equals 2 × 2 × 2 × 2 × 2 or 32).

Thus, 25x = 32x.

Note that 25x has a base of 2 and exponent of 5x.

This means that 32x can be written as an exponent that has a base of 2.

Here, we have dealt with numbers that have nice relationships with respect to the base of 2 in that they are powers of 2 (second, third, and fifth powers, respectively). But it is also possible to raise numbers to decimal powers that are irrational numbers, too.

The detailed explanation of this is beyond the scope of this book. We offer the following without proof:

(23.3219280948…)x: Using the power rule, we have

(23.3219280948…)x = 2(3.3219280948…) times x = 2(3.3219280948…)x.

But we also have that (23.3219280948…)x = (10)x = 10x (because 23.3219280949… equals 10, the demonstration of which is beyond the scope of this book).

Note that

image

Note that 2(3.3219280948…)x has a base of 2 and exponent of (3.3219280948…)x.

This means that 10x can also be written as an exponent that has a base of 2.

Observe that the number given by 3.3219280948… is a non-repeating decimal that goes on forever. We can drop the “goes on forever” and get as close an accuracy as we want by taking a sufficient number of digits. Note that if we round this off to nine digits, we have 3.321928095, and 23.321928095 ≈ 10.0000000008.

We can do this same process for an exponent to any base and thus any exponential function to any other base can be written in the form that uses 2ax. Here, a is a scenario variable or parameter. We list the various settings of a for some scenarios here:

Scenario

Value of Parameter a

2ax

4x

a = 2

22x

8x

a = 3

23x

10x

a = 3.3219280948…

2(3.3219280948…)x

16x

a = 4

24x

32x

a = 5

25x

The same reach holds true for an exponent whose base is the number e. Using the same letter for the parameter, we will have eax. Here, the number e takes the place of 2 in the previous examples. The following table lists (without proof) the various settings of a in this situation for the scenarios from the previous table:

Scenario

Value of Parameter a

eax

a Rounded to Nine Decimals

4x

a = 1.386294361…

e(1.386294361…)x

a = 1.386294361

8x

a = 2.079441541…

e(2.079441541…)x

a = 2.079441542

10x

a = 2.302585092…

e(2.302585092…)x

a = 2.302585093

16x

a = 2.772588722…

e(2.772588722…)x

a = 2.772588722

32x

a = 3.465735902…

e(3.465735902…)x

a = 3.465735903

Note that all of the a values listed here are irrational numbers, meaning they have infinite non-repeating decimal expansions and would need to be rounded off for use.

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