By definition, #pH=-log_(10)[H_3O^+]#

And thus, here, #pH=-log_(10)(10^(-10))=-(-10)=10.#

Given the definition of the logarithmic function, #log_ab=c#, #c# is that power to which we raise the base #a# to get #b#, then #log_(10)10^x=x#. Logs were introduced in the days BEFORE the advent of cheap electronic calculators; scientists, and economists, and engineers, and A level students, used to use #"log tables"#, and #"slide rules"# which made complex calculations a bit less tiresome.

In water, under standard conditions, we can show that #pH+pOH=14#. What is #[HO^-]# in the given example?

The use of #pH# and #pOH# allows the use of buffer solutions to moderate acidity and alkalinity.