Conversions : log₁₀(x) ↔ ln(x)   et   10^x ↔ exp(x)

• Relations entre 10^x et e^x :
  • ln() et exp() sont des fonctions réciproques l'une de l'autre

    ln(ex) = eln(x) = x 10 = eln(10) => 10x = [eln(10)]x = ex ln(10)

    log10(10x) = 10log10(x) = x 10log10(e) = e => ex = (10log10(e))x = 10x log10(e)

  • ou bien :

    y = 10x => ln(y) = ln(10x) = x ln(10) => eln(y) = y = 10x = ex ln(10)

    y = ex => log10(y) = log10(ex) = x log10(e) => 10log10(y) = y = ex = 10x log10(e)

  Dérivée (10^x)' = [e^{x ln(10)}]' = ln(10) [e^{x ln(10)}] = ln(10) × 10^x
   
• Relations entre log₁₀(x) et ln(x) :

  y = log10(x) => 10y = 10log10(x) = x => ln(10y) = y ln(10) = ln(x) => y = log10(x) = ln(x) / ln(10)

  y = ln(x) => ey = eln(x) = x => log10(ey) = y log10(e) = log10(x) => y = ln(x) = log10(x) / log10(e)

  Dérivée [log₁₀(x)]' = 1 / [ x ln(10) ]