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Good — we’ll do this manually (no Python) and produce a numerical table of common (base-10) logarithms for $x=300,301,\dots,310$ and then compute the first derivative of $\log_{10}x$ at $x=310$. ### Strategy / method Write each number as $x=300(1+\varepsilon)$ where $\varepsilon=\dfrac{k}{300}$ for $k=0,1,\dots,10$. Then $$ \log_{10}(300+k)=\log_{10}300+\log_{10}(1+\varepsilon) =\log_{10}3+2+\frac{\ln(1+\varepsilon)}{\ln 10}. $$ We use the series (sufficiently accurate for these small $\varepsilon$): $$ \ln(1+\varepsilon)\approx \varepsilon-\frac{\varepsilon^2}{2}+\frac{\varepsilon^3}{3}, $$ and divide by $\ln 10$ (with $\ln 10\approx 2.302585093$). (Equivalently one may use the linear approximation plus ______ _____ _____ ____ _________ _________ __________ __________ ____.
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