Question 287583
How many decimal digits are there in the number {{{2005^2005}}}
<pre><font size = 4 color = "indigo"><b>
We use this principle:

If {{{K}}} is a positive integer which when
written in its scientific notation form is:

{{{K}}}{{{"="}}}{{{M}}}{{{"×"}}}{{{10^N}}}

that is, where {{{1<=M<10}}} and {{{N}}} is a non-negative integer,

then {{{K}}} has {{{N+1}}} decimal digits.

For instance {{{8729}}}{{{"="}}}{{{8.729}}}{{{"×"}}}{{{10^3}}} has
{{{3+1}}} or {{{4}}} digits. 

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So let's get {{{2005^2005}}} in scientific notation:

Let {{{K = 2005^2005}}}

Take logs base 10 of both sides:

{{{log(10,K) = log(10,2004^2005)}}}

Use a principle of logs that allows an exponent to be written
as a coefficient:

{{{log(10,K) = 2005*log(10,2004)}}}

Use your calculator on the right sidess;

{{{log(10,K)=6620.304923}}}

Use the principle of writing a log equation as an 
equivalent exponential equation:

{{{K=10^6620.304923}}}

Write the right side as the sum of its whole part plus
its decimal part:

{{{K=10^(6620+.304923)}}}

Use the principle {{{B^(A+C)=B^A*B^C}}}

{{{K=10^6620*10^.304923}}}

Use your calculator to find {{{10^.304923}}}

{{{K=10^6620*2.018008431}}}

Write in standard scientific notation:

{{{K=2.018008431}}}{{{"×"}}}{{{10^6620}}}

Therefore K has 6621 digits, choice B.

Edwin</pre>