Find the base of x 73 (with base 8) = 214 (with base x)

Question

Find the base of x
73 (with base 8) = 214 (with base x)

Answer 1

Given what you mean is this:

\log_{8} 73 = \log_{x} 214

Then:(use change of base formula:

\log_{a} x = \frac{\ln (x)}{\ln (a)}

)

\frac{\ln (73)}{\ln (8)} = \frac{\ln (214)}{\ln (x)}

Isolate x to one side:

\frac{\ln (8)}{\ln (73)} = \frac{\ln (x)}{\ln (214)} \Rightarrow \ln (x) = \frac{\ln (8) \ln (214)}{\ln (73)}

Eliminate the logariths by rasing both sides to e:

e^{\ln (x)} = e^{\ln (\frac{\ln (8) \ln (214)}{\ln (73)})}

Solve:

x = \frac{\ln (8) \ln (214)}{\ln (73)} \Rightarrow x = 2.60070829

Answers (1)