Stumbling through mathematics: which is bigger?

Exponents are one way to represent very large numbers without having to write them out. Some examples are:

2³ = 2⋅2⋅2 = 8

9⁷ = 9⋅9⋅9⋅9⋅9⋅9⋅9⋅9⋅9 = 4782969

10¹² = 10⋅10⋅10⋅10⋅10⋅10⋅10⋅10⋅10⋅10⋅10⋅10 = 1000000000000

Now determine whether number A or number B is bigger:

A = 2^73894618

B = 3^46622313

Good luck trying to put either of those into a calculator! Microsoft Excel can't handle it either. Even Maple 12, a program specifically designed for performing algebraic calculations on a computer, tells me there are too many digits to display in either of these two numbers.

Logarithms are functions that are the inverses of exponential functions. Both logarithms and exponential functions are found in many engineering and scientific problems. The expression b = log_a(x) means b is equal to the logarithm to base a of x. Just to give you an idea of of what they look like, here is a graph of 10^x and log₁₀(x) (this plot was made using Maple 12). The red line represents 10^x and the blue line represents log₁₀(x).



Logarithms have certain limitations. The logarithm of zero does not exist, and the logarithm of a negative number is a complex number. Nevertheless, we are going to make use of the following fact concerning logarithms:

For real numbers a, x, y > 0, if x > y, then log_a(x) > log_a(y)

We will use the following property of logarithms to determine if either A or B is larger:

For real numbers z, x, a and x, a > 0, then log_a(x^z) = z⋅log_a(x)

Take the logarithm of both A and B. We will choose the common logarithm, or logarithm to base 10:

log₁₀(A) = log₁₀(2^73894618)

log₁₀(B) = log₁₀(3^46622313)

Using the property of logarithms introduced above:

log₁₀(A) = 73894618⋅log₁₀(2)

log₁₀(B) = 46622313⋅log₁₀(3)

Now we finally have something that can be evaluated! Using a calculator, we find that log₁₀(2) = 0.3010299957 (evaluated to 10 digits) and log₁₀(3) = 0.4771212549 . We now have:

log₁₀(A) = 73894618⋅0.3010299957 = 22244496.54

log₁₀(B) = 46622313⋅0.4771212549 = 22244496.48

It turns out that A is larger than B.