Learn how to solve problems step by step online. Find the limit of log(sin(2*x))/log(sin(x)) as x approaches 0. Change the logarithm to base e applying the change of base formula for logarithms: \log_b(a)=\frac{\log_x(a)}{\log_x(b)}. Change the logarithm to base e applying the change of base formula for logarithms: \log_b(a)=\frac{\log_x(a)}{\log_x(b)}. Simplify the fraction \frac{\frac{\ln\left(\sin\left(2x\right)\right)}{\ln\left(10\right)}}{\frac{\ln\left(\sin\left(x\right)\right)}{\ln\left(10\right)}}. If we directly evaluate the limit \lim_{x\to0}\left(\frac{\ln\left(\sin\left(2x\right)\right)}{\ln\left(\sin\left(x\right)\right)}\right) as x tends to 0, we can see that it gives us an indeterminate form.

Find the limit of log(sin(2*x))/log(sin(x)) as x approaches 0