The given equation can be rewritten using the properties of logarithms: 

\( \log_{10} r + \log_{10} r^2 + \log_{10} r^4 + \log_{10} r^8 + \log_{10} r^{16} + \log_{10} r^{32} = 63.\)

Using the property \( \log_{a} b^n = n \log_{a} b \), we can simplify each term: \( \log_{10} r + 2 \log_{10} r + 4 \log_{10} r + 8 \log_{10} r + 16 \log_{10} r + 32 \log_{10} r = 63 \)

Combine the logarithms using the properties of logarithms: \( \log_{10} (r \cdot r^2 \cdot r^4 \cdot r^8 \cdot r^{16} \cdot r^{32}) = 63 \) 

Simplify the expression inside the logarithm: 

\( \log_{10} r^{63} = 63 \( Since \( \log_{10} r^{63} = 63 \), this implies that \( r^{63} = 10^{63} \). To solve for \( r \), take the 63rd root of both sides: \( r = \sqrt[63]{10^{63}} = 10^1 = 10 \)