6.7Exponential and Logarithmic Equations

‘Estimating Temperature (or Vapor Pressure)’ solves the Clausius-Clapeyron equation for T2 by isolating 1/T2 on one side and then raising the result to the −1 power to undo the reciprocal — the same isolate-then-invert moves used to solve any exponential/logarithmic equation for a variable trapped inside both a log and a reciprocal.

ΔG = ΔG° + RT ln Q and its equilibrium special case ΔG° = −RT ln K, K = e^(−ΔG°/RT), require solving an equation that mixes a variable inside and outside a natural-log/exponential — exactly an exponential-and-logarithmic-equation solve, used here to compute Ksp for AgCl from tabulated free energies of formation.

Going from pH back to [H3O+] requires solving pH = −log[H3O+] for the concentration, i.e. exponentiating both sides to get [H3O+] = 10^(−pH) — the blood-pH example calls this out explicitly as taking the antilog, or the inverse log.

Computing percent ionization from a given pH first requires converting pH to [H3O+] = 10^(−pH) — the same exponential-equation move as pOH↔pH conversion, chained into a rational-expression calculation.

Getting K from E°cell = (RT/nF) ln K requires isolating K by exponentiating both sides — worked here as K = 10^(nE°cell/0.0592) — the same exponential-equation move used for ΔG°/K conversions, now driven by a measured cell potential instead of a tabulated free energy.

The whole method of initial rates, once ratios stop being clean integers, reduces to solving aᵐ = b for the unknown exponent m by taking logs of both sides — the exponential-equation technique itself; eyeballing that doubling [NO] doubles the rate works for m = 1 but breaks the moment a ratio like 2.25 or a fractional order appears.

Every half-life and time-to-decompose problem (‘how long until 80% has decomposed?’) is solved by isolating t in [A]ₜ = [A]₀e^(−kt) via natural log — the exponential-equation-solving skill itself; a student who cannot move between the exponential and log forms cannot finish these problems even though the chemistry setup (x and 0.200x) is trivial.

Cobalt-60 dosimetry, radiocarbon dating, and uranium-lead rock dating are the same algebra problem in different units: isolate t in Nₜ = N₀e^(−λt) by taking a natural log of both sides; a student who cannot perform that step cannot finish a single worked example in this section no matter how well they understand half-lives conceptually.