To solve this logarithm problem, we can use the basic rules of logarithms to rewrite the equation as n^6 = log_2 (n) + log_n (486√2). Then, we can use the change of base formula to convert the logarithms to a common base, such as base 10 or base e, and solve for n.
The value of n in this equation is approximately 2.077. This can be found by solving the equation n^6 = log_2 (n) + log_n (486√2) using a calculator or algebraic manipulation.
Using different bases in this equation allows us to manipulate the logarithms and make them easier to solve. By converting the logarithms to a common base, we can use basic algebra to solve for n.
Yes, this equation can be solved without using logarithms. We can rewrite the equation as n^6 = n^(log_2 (1944)) = n^(log_n (486√2)) and use the fact that two exponential expressions with the same base are equal if and only if their exponents are equal. This will result in a simpler equation that can be solved using basic algebra.
To check if your answer is correct, you can substitute your value of n back into the original equation and see if it satisfies the equation. You can also use a calculator to evaluate both sides of the equation separately and see if they are equal.