Problem 85 - Algebra and Trigonometry

Question

The Googolplex A googol is \(10^{100},\) and a googolplex is \(10^{\text { googol }} .\) Find \(\log (\log (\operatorname{googol}))\) and \(\quad \log (\log (\log (\text { googolplex })))\)

Step-by-Step Solution

Answer

Both \(\log(\log(\text{googol}))\) and \(\log(\log(\log(\text{googolplex})))\) are 2.

Step 1: Understanding Logarithm of a Googol

A googol, denoted as \(10^{100}\), can be simplified when we take the logarithm. Since the logarithm of a power simplifies as: \(\log(a^b) = b \cdot \log(a)\), we find \(\log(\text{googol}) = \log(10^{100}) = 100 \cdot \log(10)\). Since \(\log(10) = 1\), we have \(\log(10^{100}) = 100\).

Step 2: Finding Logarithm of Logarithm of a Googol

Now, calculate \(\log(\log(\text{googol}))\). We have determined that \(\log(\text{googol}) = 100\). Thus, \(\log(\log(\text{googol})) = \log(100)\). Using the power of ten representation, \(100 = 10^2\), so \(\log(100) = 2\cdot\log(10) = 2\). Thus, \(\log(\log(\text{googol})) = 2\).

Step 3: Understanding Logarithm of a Googolplex

A googolplex is \(10^{\text{googol}} = 10^{10^{100}}\). We can start by calculating the logarithm: \(\log(\text{googolplex}) = \log(10^{10^{100}}) = 10^{100} \cdot \log(10)\). Since \(\log(10) = 1\), we get \(\log(10^{10^{100}}) = 10^{100}\).

Step 4: Finding Logarithm of Logarithm of a Googolplex

We need to compute \(\log(\log(\text{googolplex}))\). From the previous step, \(\log(\text{googolplex}) = 10^{100}\), thus, \(\log(\log(\text{googolplex})) = \log(10^{100}) = 100\cdot \log(10) = 100\).

Step 5: Finding Logarithm of Logarithm of Logarithm of a Googolplex

Now, calculate \(\log(\log(\log(\text{googolplex})))\). We found that \(\log(\log(\text{googolplex})) = 100\). Thus, \(\log(100)\) is computed similarly as before, resulting in \(\log(100) = 2\). Therefore, \(\log(\log(\log(\text{googolplex}))) = 2\).

Key Concepts

Googol

A googol is an incredibly large number, defined as 10 raised to the power of 100, and is written as \(10^{100}\). This concept might sound abstract or even humorous, but it serves a purpose in mathematics to illustrate very large numbers. To give some perspective, a googol is much larger than the number of atoms in the observable universe, which is estimated to be about \(10^{80}\).

The term googol was coined by a nine-year-old named Milton Sirotta, who was the nephew of the mathematician Edward Kasner. The idea wasn't just to introduce an advanced concept, but to help with understanding one aspect of exponentiation — showing how rapidly numbers increase when written as powers.

• A googol is \(10^{100}\).
• Useful in discussions about scale and size in mathematics.

Exploring a googol helps us appreciate the concept of numbers far beyond the regular scales we use every day.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, the base, to the power of an exponent. It is a way to express repeated multiplication of the same number. For example, \(10^3\) means multiplying 10 by itself three times: \(10 \times 10 \times 10\).

Exponentiation is a fundamental part of mathematics, used in various fields to simplify expressions and calculations. It is integral to understanding large numbers and scientific notation.

• The base is the number being multiplied.
• The exponent tells you how many times to multiply the base by itself.
• Examples: \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).

The power of exponentiation is clearly seen in the concepts of googol and googolplex, showcasing its ability to handle and represent extremely large numbers efficiently.

Power of Ten

The power of ten is a simplified way to express very large or very small numbers by using a base of ten with an exponent. Numbers like 100, 1000, or even 0.01 are each powers of ten. The number 100 can be written as \(10^2\), and 0.01 can be written as \(10^{-2}\). Working with powers of ten makes it easy to comprehend and perform calculations with large and small quantities.

The concept is especially handy in scientific notation, which is commonly used to render extremely large numbers manageable. A googol (\(10^{100}\)) and a googolplex (\(10^{10^{100}}\)) both rely on powers of ten for their expression.

• Expressing numbers as powers of ten simplifies calculations.
• Scientific notation facilitates working with large and small numbers.

Learning how to interpret and use powers of ten is crucial for students tackling topics involving massive numbers, like astronomy and quantum physics.

Logarithmic Functions

Logarithmic functions are the inverse of exponentiation. They help us solve problems involving exponential growth or decay by "undoing" the exponentiation process. When you take the logarithm of a number, you're asking "what power do we need to raise the base, usually 10, to get this number?" For example, \(\log_{10}(100) = 2\), because \(10^2 = 100\).

Logarithmic functions streamline the resolution of complex equations, especially in cases of very large numbers, like googol, or very small numbers. They are fundamental in fields like biology, finance, and engineering.

• The common log base is 10, noted as \(\log_{10}\).
• The natural log base is Euler’s number, \(e\), noted as \(\ln\).

By using logarithms, we can manage and interpret data involving exponential growth or geometric progression effectively, providing us with concise solutions to otherwise convoluted calculations.