Formula for moving to a new foundation

Change of base

Let $a>0$ , $a\neq1$ , $b>0$ , $c>0$ , $c\neq1$ . Then the logarithm of $b$ to base $a$ can be written using a logarithm to any admissible base $c$ : $\log_a b=\dfrac{\log_c b}{\log_c a}$ . People often take $c=10$ ( $\lg$ ) or $c=e$ ( $\ln$ ) to compute on a calculator or to speak in one “language”. Denominator is nonzero: under our assumptions $\log_c a\neq0$ (otherwise $a=1$ , which is excluded).

Numerator and denominator are logarithms to the same new base $c$

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