Actually, IIRC, gravitational field inside a solid sphere increases linearly with r for r less than R and drops off with 1/r^2 for r greater than R. In this case the field would be 50% of the field at the surface. I don't know where you got "logarithmic" from.
Of course this is assuming the earth is a perfect sphere of uniform density--if anyone has an even more accurate calculation, feel free to add it too
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