What is the number of digits in 2 40 ? (Given that log 10 2 = 0.301)
13
Determining the Number of Digits in an Exponential Expression
To find the number of digits in a positive integer \(N\), we can use logarithms. Specifically, the number of digits in \(N\) is given by the formula:
Number of digits = \( \lfloor \log_{10} N \rfloor + 1 \)
Here, \( \lfloor x \rfloor \) represents the greatest integer less than or equal to \(x\) (the floor function).
Calculating the Logarithm of 240
In this problem, we need to find the number of digits in the number \( 2^{40} \). So, we set \( N = 2^{40} \). We are given the value of \( \log_{10} 2 \). Let's apply the formula:
- First, find the base-10 logarithm of \( 2^{40} \): \[ \log_{10} (2^{40}) \]
- Using the power rule of logarithms, which states that \( \log_b (x^y) = y \cdot \log_b x \), we can rewrite the expression: \[ \log_{10} (2^{40}) = 40 \times \log_{10} 2 \]
- We are given that \( \log_{10} 2 = 0.301 \). Substitute this value into the equation: \[ 40 \times 0.301 \]
- Perform the multiplication: \[ 40 \times 0.301 = 12.04 \]
- Now, apply the formula for the number of digits using the result \( \log_{10} (2^{40}) = 12.04 \): Number of digits = \( \lfloor \log_{10} (2^{40}) \rfloor + 1 \) Number of digits = \( \lfloor 12.04 \rfloor + 1 \)
- Calculate the floor of \( 12.04 \), which is the greatest integer less than or equal to \( 12.04 \): \( \lfloor 12.04 \rfloor = 12 \)
- Finally, add 1 to the result: Number of digits = \( 12 + 1 = 13 \)
Conclusion
Therefore, the number of digits in \( 2^{40} \) is 13.
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