pH of pure water
- Input
- x = 1e-7
- Expected output
- log₁₀(1×10⁻⁷) = -7
pH = −(−7) = 7, the scale's neutral reference value.
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log base 10 calculator
log₁₀(x) is the exponent 10 must be raised to in order to reach x: log₁₀(100)=2 because 10²=100. It is the base used by definition for pH, decibels and earthquake magnitude, because it compresses thousand-fold variations into a scale of just a few numbers.
pH = −(−7) = 7, the scale's neutral reference value.
100 is exactly 10², and the exponent 2 also signals the number has 3 digits (2+1).
That is why 2¹⁰=1024 has 4 digits: ⌊10×0.30103⌋+1=4, matching 1024's real digit count.
log_b(x) = y means b^y = x, so a logarithm is simply an exponent, the power the base must be raised to. It is the inverse operation of exponentiation. For example, log₂(8) = 3 because 2³ = 8, and log₁₀(100) = 2 because 10² = 100.
Through pH=−log₁₀[H⁺]. For pure water, [H⁺]=1×10⁻⁷ mol/L, log₁₀ of that is −7, and pH comes out to 7. Each pH unit lower multiplies the H⁺ concentration by 10, so pH 4 is 1000 times more acidic than pH 7.
For positive integers, yes: digit count equals ⌊log₁₀(x)⌋+1. Since log₁₀(100)=2 exactly, 100 has 2+1=3 digits; for 2¹⁰=1024, log₁₀(1024)≈3.0103, giving ⌊3.0103⌋+1=4 digits, the correct value.
Because sound power and seismic energy vary across enormous orders of magnitude; log₁₀ compresses that range into a small scale, where 10 more dB means 10 times more power, and one more Richter degree means 10 times more wave amplitude.
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