Why do we use e as the base of natural logarithms instead of 10?

The base $e$ is used because it is the unique base for which an exponential's steepness at every point equals its height at that point, so differentiating or integrating continuous growth introduces no extra constant. Base ten would give the same logarithm up to a fixed multiplier, but every derivative would carry a factor of $\ln 10$, about 2.3026.

Take $b^x$ and ask how steep it is where $x=0$. For $b=10$ the slope is $\ln 10$, so near zero $10^x$ changes by about $2.3026$ times the change in $x$. For $b=2$ the slope is $\ln 2$, about $0.6931$. For $b=e$ the slope is exactly 1. That single condition, steepness equal to height, is what calculus asks of an exponential, and only $e$ satisfies it at every point: $d(e^x)/dx=e^x$. The same fact read backwards is why $\ln x$ has derivative $1/x$ while every other logarithm has a constant in front. The notation $\ln$ means $\log_e$. That is the whole reason the natural logarithm is the one that appears in integrals as well as derivatives.

Changing base is one division: $\log_{10}x=\ln x/\ln10$. So the base ten logarithm is the natural logarithm scaled by $1/\ln10$, which is about 0.4343. Differentiating gives $0.4343/x$, not $1/x$. For example, $\log_{10}100=2$ and $\ln100=4.605170$, so dividing by $\ln10=2.302585$ returns 2. That division is the whole change of base formula. In a differential equation the constant does not vanish. A quantity growing at a rate proportional to its size satisfies $y=y_0e^{kt}$; the same solution in base ten is $y=y_0 10^{(k/\ln10)t}$. It is correct, but the exponent now contains $k/\ln10$ where the natural version contains $k$. Every integration by parts, every gradient, every maximum likelihood derivative carries that factor if the base is ten.

All logarithms are proportional, so base ten is exactly as good as base $e$ for comparing sizes, taking ratios, or saying that one quantity grows faster than another. That is the part the natural guess gets right. It fails at the moment a derivative or an integral appears. A constant multiplier cancels in a ratio but not in a gradient, so the conversion factor stops being bookkeeping and becomes part of the rate. Base ten is not wrong; it is a unit with a tax of $\ln10$ on every calculus operation. Base $e$ is the unit where that tax is zero.

Insisting on base ten breaks nothing. It merely moves the awkward constant from the notation into the working. The clean statements, such as the derivative of the logarithm being $1/x$ and the solution of $dy/dt=ky$ being $y_0e^{kt}$, are clean precisely because the base was chosen to make the constant one. Base $e$ is not more true than base ten; it is the base that leaves the calculus of growth unadorned. Change the base and every conclusion about proportion survives, because the conversion factor cancels in every ratio.