Uncertainty in the output when 10 is raised to the power of a value with an associated uncertainty?

Question

For the purpose of error propagation, I need to raise 10 to the power of a value with an uncertainty. How would this error be propagated to the new 'out' value? Thanks in advance.

Answer 1

According to Section 3.7 of An Introduction to Error Analysis: the Study of Uncertainties in Physical Measurements, Second Edition by John R. Taylor,

If $x$ is measured with uncertainty $\delta x$ and is used to calculate the function $q(x)$, then the uncertainty $\delta q$ is $$\delta q = \left| \frac{dq}{dx} \right| \delta x$$

If $q(x) = 10^x$, then $q'(x) = \ln(10)\cdot 10^x$, so $\delta q = \ln(10) \cdot 10^x \delta x$.