Find the limit of log2(x)/logc(x) as x approaches 0

 Exercise

$\lim_{x\to0}\left(\frac{\log2\left(x\right)}{\log_c\left(x\right)}\right)$

 

Change the logarithm to base $e$ applying the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

$\lim_{x\to0}\left(\frac{\frac{\ln\left(x\right)}{\ln\left(2\right)}}{\log_{c}\left(x\right)}\right)$

 

Change the logarithm to base $e$ applying the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

$\lim_{x\to0}\left(\frac{\frac{\ln\left(x\right)}{\ln\left(2\right)}}{\frac{\ln\left(x\right)}{\ln\left(c\right)}}\right)$

 

Simplify the fraction $\frac{\frac{\ln\left(x\right)}{\ln\left(2\right)}}{\frac{\ln\left(x\right)}{\ln\left(c\right)}}$

$\lim_{x\to0}\left(\frac{\ln\left(c\right)}{\ln\left(2\right)}\right)$

 

The limit of a constant is just the constant

$\frac{\ln\left(c\right)}{\ln\left(2\right)}$

$\frac{\ln\left(c\right)}{\ln\left(2\right)}$

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