 # Find the base of x 73 (with base 8) = 214 (with base x) Kenya Leonard 2022-08-01 Answered
Find the base of x
73 (with base 8) = 214 (with base x)
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Given what you mean is this:
${\mathrm{log}}_{8}73={\mathrm{log}}_{x}214$
Then:(use change of base formula: ${\mathrm{log}}_{a}x=\frac{\mathrm{ln}\left(x\right)}{\mathrm{ln}\left(a\right)}$)
$\frac{\mathrm{ln}\left(73\right)}{\mathrm{ln}\left(8\right)}=\frac{\mathrm{ln}\left(214\right)}{\mathrm{ln}\left(x\right)}$
Isolate x to one side:
$\frac{\mathrm{ln}\left(8\right)}{\mathrm{ln}\left(73\right)}=\frac{\mathrm{ln}\left(x\right)}{\mathrm{ln}\left(214\right)}⇒\mathrm{ln}\left(x\right)=\frac{\mathrm{ln}\left(8\right)\mathrm{ln}\left(214\right)}{\mathrm{ln}\left(73\right)}$
Eliminate the logariths by rasing both sides to e:
${e}^{\mathrm{ln}\left(x\right)}={e}^{\mathrm{ln}\left(\frac{\mathrm{ln}\left(8\right)\mathrm{ln}\left(214\right)}{\mathrm{ln}\left(73\right)}\right)}$
Solve:
$x=\frac{\mathrm{ln}\left(8\right)\mathrm{ln}\left(214\right)}{\mathrm{ln}\left(73\right)}⇒x=2.60070829$