How high the tower?

Algebra Level 3

The number googol is defined as a 1 with 100 zeros after it. As an exponent: $10^{100}$ .

A googolplex is defined as a 1 with a googol zeros after it. As an exponent: $10^{10^{100}}$ or $10^{10^{10^{2}}}$ .

Since $10^{0.301}=2$ this can even be written as $\huge 10^{10^{10^{10^{0.301}}}}$ . This is a tower of four 10's with a decimal between zero and one at the top.

Write googolplex instead as a tower of $\large e$ 's with a decimal between zero and one at the top.

Let $A$ =the number of $\large e$ 's required and $B$ = the decimal at the top. Give the sum $A+B$

The answer is 5.52426.

1 solution

Jeremy Galvagni
Jul 28, 2018

Basically, you need to count how many natural logarithms will bring the googolplex down below 1. I will use $\ln$ for the natural logarithm.

1st ln: $\ln{10^{googol}}=googol \cdot \ln{10}$

2nd ln: $\ln{googol \cdot \ln{10}}= 100 \ln{10}+ \ln {\ln{10}}$ which is small enough to evaluate with a calculator $231.0925417$

3rd ln: $\ln{231.0925417}=5.442818244$

4th ln: $\ln{5.442818244}=1.694296986$

5th ln: $\ln{1.694296986}=0.5272678974$

So $A=5$ and $B=0.5272678974$ . $A+B= \boxed{5.5272678974}$

Last line is A+B=,not A+B+

X X - 2 years, 10 months ago

Fixed, thanks! Now to figure out how to fix the typo in the solution. Fortunately, it's below the error threshold for the actual correct answer.

Jeremy Galvagni - 2 years, 10 months ago

How high the tower?

The answer is 5.52426.

1 solution

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