How Many Digits Does a Large Exponent Have?
Calculating the Number of Digits in ab
For small exponential expressions such as 58, you can determine its number of digits simply by evaluating the expression and counting the digits. For instance, 58 = 390625, an integer with 6 digits.
However, it is more challenging to count the number of digits in a large exponent such as 4710000. If you try to evaluate it on a standard calculator, you will get an overflow error message since it is a number with 16721 digits.
Using Log10 to Calculate the Number of DigitsThe formula for the number of digits D in ab is
D = 1 + Log10(ab) = 1 + (b)Log10(a),
where ⌊ ⌋ is the floor function. Logarithm base-10 appears in this formula because we use a base-10 number system.
To see why the above formula works, consider the smallest 4-digit number and the smallest 5-digit number--1000 and 10000. Notice that 1000 = 103 and has 4 digits, and Log101000 = 3. Similarly, 10000 = 104 and has 5 digits, and Log1010000 = 4. The number of digits in 1000 and 10000 is one more than their logarithms.
Now consider a number between 1000 and 10000, for example, 5481. This is a number with 4 digits and Log105481 = 3.7389. The number of digits in 5481 is equal to 3.7389 rounded up to the nearest whole number.
Thus, to compute D, you take the logarithm of the number in base-10, add 1, and then round down to the nearest integer.
Example: Calculate the number of digits in 77778888.
D = 1 + Log1077778888 = 1 + 8888Log107777 = 34582.
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