Modulus of Change of Base of Logarithm/Examples

Examples of Modulus of Change of Base of Logarithm

Base $e$ to Base $10$

Consider the change from base $e$ to base $10$:

$\log_{10} x = \paren {\log_{10} e} \paren {\ln x} = 0 \cdotp 43429 \, 44819 \, 03 \ldots \ln x$

Then $\log_{10} e = 0 \cdotp 43429 \ldots$ is the modulus.

Base $10$ to Base $e$

Consider the change from base $10$ to base $e$:

$\ln x = \paren {\ln 10} \paren {\log_{10} x} = 2 \cdotp 30258 \, 50929 \, 94 \ldots \log_{10} x$

Then $\log_e 10 = 2 \cdotp 30258 \, 5 \ldots$ is the modulus.