Appendix 4
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.
• (2^{3})^{2}: This means to raise 2^{3} to the second power, so we have a power raised to another power.
Doing so yields (2^{3})^{2} = (2^{3})(2^{3}).
This equals (2 × 2 × 2)(2 × 2 × 2) because 2^{3} means to multiply three 2s together.
Thus, (2^{3})^{2} = (2 × 2 × 2)(2 × 2 × 2).
Dropping parentheses gives
This means that (2^{3})^{2} is the equivalent of multiplying six 2s together.
Thus, (2^{3})^{2} equals 2^{6} or (2^{3})^{2} = 2^{3×2}. (This is the power rule.)
• (5^{4})^{3}: This means to raise 5^{4} to the third power.
Doing so yields (5^{4})^{3} = (5^{4})(5^{4})(5^{4}).
This equals (5 × 5 × 5 × 5)(5 × 5 × 5 × 5)(5 × 5 × 5 × 5) because 5^{4} means to multiply four 5s together.
Thus, (5^{4})^{3} = (5 × 5 × 5 × 5)(5 × 5 × 5 × 5)(5 × 5 × 5 × 5).
Dropping parentheses gives
This means that (5^{4})^{3} is the equivalent of multiplying 12 5s together.
Thus, (5^{4})^{3} equals 5^{12} or (5^{4})^{3} = 5^{4×3}. (This is the power rule.)
• (2^{2})^{x}: Using the power rule, we have (2^{2})^{x} = 2^{2 · x} = 2^{2x}.
But we also know that (2^{2})^{x} = (4)^{x} = 4^{x} (because 2^{2} equals 4).
Thus, 2^{2x} = 4^{x}.
Note that 2^{2x} has a base of 2 and exponent of 2x.
This means that 4^{x} can be written as an exponent that has a base of 2.
• (2^{3})^{x}: Using the power rule, we have (2^{3})^{x} = 2^{3 · x} = 2^{3x}.
But we also know that (2^{3})^{x} = (8)^{x} = 8^{x} (because 2^{3} equals 8).
Thus, 2^{3x} = 8^{x}.
Note that 2^{3x} has a base of 2 and exponent of 3x.
This means that 8^{x} can be written as an exponent that has a base of 2.
• (2^{5})^{x}: Using the power rule, we have (2^{5})^{x} = 2^{5 · x} = 2^{5x}.
But we also know that (2^{5})^{x} = (32)^{x} = 32^{x} (because 2^{5} equals 2 × 2 × 2 × 2 × 2 or 32).
Thus, 2^{5x} = 32^{x}.
Note that 2^{5x} has a base of 2 and exponent of 5x.
This means that 32^{x} 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:
• (2^{3.3219280948…})^{x}: Using the power rule, we have
(2^{3.3219280948…})^{x} = 2^{(3.3219280948…) times x} = 2^{(3.3219280948…)x}.
But we also have that (2^{3.3219280948…})^{x} = (10)^{x} = 10^{x} (because 2^{3.3219280949…} equals 10, the demonstration of which is beyond the scope of this book).
Note that
Note that 2^{(3.3219280948…)x} has a base of 2 and exponent of (3.3219280948…)x.
This means that 10^{x} 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 2^{3.321928095} ≈ 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 2^{ax}. Here, a is a scenario variable or parameter. We list the various settings of a for some scenarios here:
Scenario |
Value of Parameter a |
2^{ax} |
4^{x} |
a = 2 |
2^{2x} |
8^{x} |
a = 3 |
2^{3x} |
10^{x} |
a = 3.3219280948… |
2^{(3.3219280948…)x} |
16^{x} |
a = 4 |
2^{4x} |
32^{x} |
a = 5 |
2^{5x} |
The same reach holds true for an exponent whose base is the number e. Using the same letter for the parameter, we will have e^{ax}. 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 |
e^{ax} |
a Rounded to Nine Decimals |
4^{x} |
a = 1.386294361… |
e^{(1.386294361…)x} |
a = 1.386294361 |
8^{x} |
a = 2.079441541… |
e^{(2.079441541…)x} |
a = 2.079441542 |
10^{x} |
a = 2.302585092… |
e^{(2.302585092…)x} |
a = 2.302585093 |
16^{x} |
a = 2.772588722… |
e^{(2.772588722…)x} |
a = 2.772588722 |
32^{x} |
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.