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# How Many Digits Does a Large Exponent Have?

## Calculating the Number of Digits in a^{b}

For small exponential expressions such as 5^{8}, you can determine its number of digits simply by evaluating the expression and counting the digits. For instance, 5^{8} = 390625, an integer with 6 digits.

However, it is more challenging to count the number of digits in a large exponent such as 47^{10000}. 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.

*count*the number of digits in a larger exponent, it is easier to

*calculate*their number. You can do this using logarithms and the explanation below, or the calculator at left.

### Using Log_{10} to Calculate the Number of Digits

The formula for the number of digits D in a^{b}is

D = 1 + Log

_{10}(a

^{b}) = 1 + (b)Log

_{10}(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 = 10

^{3}and has 4 digits, and Log

_{10}1000 = 3. Similarly, 10000 = 10

^{4}and has 5 digits, and Log

_{10}10000 = 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 Log

_{10}5481 = 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 7777

^{8888}.

D = 1 + Log

_{10}7777

^{8888}= 1 + 8888Log

_{10}7777 = 34582.

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