<PRE><B>Question 43039</B>
&lt;pre&gt;&lt;font size = 5&gt;&lt;b&gt;Given that 0.475 &lt; log base 10 of 3 &lt; 0.478, how 
many digits are in the number 3&lt;sup&gt;50&lt;/sup&gt;? 

We have to get 3&lt;sup&gt;50&lt;/sup&gt; into the form 10&lt;sup&gt;x&lt;/sup&gt;

We need to find x such that

3&lt;sup&gt;50&lt;/sup&gt; = 10&lt;sup&gt;x&lt;/sup&gt;

Take logs (base 10 understood) of both sides

log(3&lt;sup&gt;50&lt;/sup&gt;) = log(10&lt;sup&gt;x&lt;/sup&gt;)

Using rules of logarithms:

50·log(3) = x

Solve for log(3)

log(3) = x/50

We are given that

0.475 &lt; log(3) &lt; 0.478

So substitute x/50 for log(3)

0.475 &lt; x/50 &lt; 0.478

Multiply thru by 50

23.75 &lt; x &lt; 23.9

Raise 10 to all three powers:

10&lt;sup&gt;23.75&lt;/sup&gt; &lt; 10&lt;sup&gt;x&lt;/sup&gt; &lt; 10&lt;sup&gt;23.9&lt;/sup&gt;

10&lt;sup&gt;23.75&lt;/sup&gt; &lt; 3&lt;sup&gt;50&lt;/sup&gt; &lt; 10&lt;sup&gt;23.9&lt;/sup&gt;

Integers equal to or greater than 10&lt;sup&gt;0&lt;/sup&gt; but less than 10&lt;sup&gt;1&lt;/sup&gt;
have 1 digit.
Integers equal to or greater than 10&lt;sup&gt;1&lt;/sup&gt; but less than 10&lt;sup&gt;2&lt;/sup&gt;
have 2 digits.
Integers equal to or greater than 10&lt;sup&gt;2&lt;/sup&gt; but less than 10&lt;sup&gt;3&lt;/sup&gt;
have 3 digits.
Integers equal to or greater than 10&lt;sup&gt;3&lt;/sup&gt; but less than 10&lt;sup&gt;4&lt;/sup&gt;
have 4 digits.
...      ...      ...      ...      ...      ...      ...      ...
Integers equal to or greater than 10&lt;sup&gt;23&lt;/sup&gt; but less than 10&lt;sup&gt;24&lt;/sup&gt;
have 24 digits.

10&lt;sup&gt;23&lt;/sup&gt; &lt; 10&lt;sup&gt;23.75&lt;/sup&gt; &lt; 3&lt;sup&gt;50&lt;/sup&gt; &lt; 10&lt;sup&gt;23.9&lt;/sup&gt; &lt; 10&lt;sup&gt;24&lt;/sup&gt;

Thus 3&lt;sup&gt;50&lt;/sup&gt; has 24 digits.

Checking with the calculator: 3&lt;sup&gt;50&lt;/sup&gt; = 7.176879877×10&lt;sup&gt;23&lt;/sup&gt;


Edwin
AnlytcPhil@aol.com&lt;/pre&gt;</PRE>