log a 0,5 = 3 a=?

Frage:  log(0.5) zur Basis a = 3    Wie groß ist  a ?  Ist das so gemeint ?

$${\frac{{log}_{10}\left({\mathtt{0.5}}\right)}{{log}_{10}\left({\mathtt{a}}\right)}} = {\mathtt{3}}$$       =>    $${\mathtt{3}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{a}}\right) = {log}_{10}\left({\mathtt{0.5}}\right)$$

$${\mathtt{3}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{a}}\right) = {log}_{10}\left({\mathtt{0.5}}\right) \Rightarrow {\mathtt{a}} = {{\mathtt{e}}}^{{\mathtt{\,-\,}}\left({\frac{{\mathtt{13\,614\,799}}}{{\mathtt{58\,926\,009}}}}\right)} \Rightarrow {\mathtt{a}} = {\mathtt{0.793\: \!700\: \!525\: \!984\: \!099\: \!6}}$$

Probe:     $${\frac{{log}_{10}\left({\mathtt{0.5}}\right)}{{log}_{10}\left({\mathtt{0.793\: \!7}}\right)}} = {\mathtt{2.999\: \!991\: \!395\: \!376\: \!062\: \!2}}$$

Gruß radix !