math log question

Question

pH = 8.34 \[H3O+\] = 10\^-8.34 = 4.57 x 10-9 M ​ i suck at math. can someone explain how they got from 10\^-8.34 to 4.57 x 10-9 M

Answer 1

If you're taking chemistry, and you're encountering logarithms, you're familiar with exponents. Logarithms are the inverse (mathematically opposite) operations of exponents. We can write an exponential expression like 2³ = 8 as a logarithm too: log₂ 8 = 3. It works like this: the base of the log raised to the answer equals the number you're taking the log of. A "standard logarithm" has a base 10, so without adding a base, we can say that log 100 = 2, since 10² = 100.

In your calculations about pH, 10^-8.34 = 4.57 with the decimal moved nine places to the left, or in scientific notation, 4.57(10^-9). Writing this in the form of a standard logarithm, with base 10, instead of 10^-8.34 = 4.57(10^-9), we get log (4.57x10^-9) = -8.34. The two expressions are equivalent. I'm sure you know we use exponents to write really big numbers that would be a pain (or impossible) to write otherwise. We use logarithms to write really small numbers, like this one with nine zeros in front of the significant digits.

I hope that helps!