Error propagation with Log base 10 I have some data and some errors for the data. The task I have been given is to $\log_{10}$ both sets of data, and then deal with the errors to plot a graph. For natural logarithms, this is easier as (I believe) you can use $\sigma_f = \sigma_x / x$ for $f = \ln(x)$ however I am struggling to find any formula/guidance on how to propagate the errors for none-natural logs.
 A: As you seem to already know, if you have a function such as $$f(x) = \ln{x},$$ then we can simply differentiate the above equation to get that: $$\text{d}f = \frac{\text{d}x}{x} \quad \quad \implies \quad \quad \Delta f \approx \frac{\Delta x}{x},$$ where in the last step I have assumed that the errors are sufficiently small so that the differentials $\text{d}f$ and $\text{d}x$ can be safely "replaced" by the absolute errors $\Delta f$ and $\Delta x$ respectively. (Such a calculus-based approach to error-analysis is useful, and you can read up more about it here.)
Now, if you instead have $$f(x) = \log_{10}(x),$$ the method is identical, the only thing you need to know is a standard result from logarithms, that: $$\log_{10}(x) = \frac{\ln(x)}{\ln{10}}.$$ As a result, $$\Delta f = \frac{1}{\ln 10}\frac{\Delta x}{x}.$$

As @rob points out in the comments below, it is important to stress that this holds true only when $\Delta x\ll x$. If this is no longer true, then the error bars will be asymmetric, and different methods will have to be employed to deal with individual situations. As pointed out in @EmilioPisanty's answer here, if $\Delta x = x$ (in a "worst case" scenario), then the value of $x$ could be anything between $0$ and $2x$, meaning that the error in $\ln(x)$ could be anything between $-\infty$ and $\ln(2x)$, which is clearly not symmetric about the point $\ln(x)$!
