I'm currently 11th grader in secondary school and we are learning logarithms. I think I know them quite well but I'm stuck on this exercise:
Evaluate $2^x - 10^y$, where $x=1/log_6(2)$ and $y=2/log_2(10)$.
When I plug the values in the calculator I get an answer of 10. But can someone show me the algorithm on how to calculate this? Because I'm unable to get the answer by myself (Sorry if some terms are not on point, I'm not a native English speaker.)
Given $$\displaystyle x= \frac{1}{\log_{6}(2)} = \log_{2}(6)$$ and $$\displaystyle y = \frac{2}{\log_{2}(10)} = 2\log_{10}(2) = \log_{10}(2)^2 = \log_{10}(4)$$
Now $$2^x-10^y = 2^{\log_{2}(6)} -10^{\log_{10}(4)} = 6-4=2$$
Above we have used the formula $$\displaystyle \bullet\; \log_{a}(b)\times \log_{b}(a) =1$$ and $$\displaystyle \bullet\; n\times \log_{e}(m) = \log_{e}(m)^n$$
and $$\displaystyle \bullet\; a^{\log_{a}(x)} = x$$