Question 838174
the solving of this problem involves the use of logarithms since the variable is in the exponent.


start with:


16^(2-n) = (1/4)^(n+1)


take the log of both sides of the equation to get:


log((16)^(2-n)) = log((1/4)^(n+1))


by the properties of logarithms, log(a^b) = b * log(a), your equation becomes:


(2-n) * log(16) = (n+1) * log(1/4)


divide both sides of this equation by log(16) to get:


2-n = (n+1) * log(1/4) / log(16).


find the value of log(1/4) / log(16) and replace in your equation to get:


2-n = (n+1) * (-.5)


simplify by distributing the multiplication on the right side of this equation to get:


2-n = -.5 * n - .5


add .5 * n to both sides of this equation to get:


2 - n + .5n = -.5


subtract 2 from both sides of this equation to get:


-n + .5n = -.5 - 2


simplify by combining like terms to get:


-.5n = -2.5


divide both sides of this equation by -.5 to get


n = 5


that should be your answer.


confirm by substituting in your original equation to see if the equation is true.


your original equation is:


16^(2-n) = (1/4)^(n+1)


replace n with 5 to get:


16^(2-5) = (1/4)^(5+1)


simplify to get:


16^(-3) = (1/4)^6


simplify further to get:


.000244140625 = .000244140625


this confirms the solution is correct.


the value of n is 5.


note that the method used to solve is the same whether the equation is complex or not.


it's the arithmetic that can get a little hairy when the problem is complicated.


your simple example was:


128^n = 8 


you would take the log of both sides of this equation to get:


log(128^n) = log(8)


you would then transform using the properties of logs to get:


n * log(128) = log(8)


you would then divide both sides of this equation by log(128) to get:


n = log(8) / log(128)


the complex problem is done in a similar manner but the arithmetic became more complicated and it required more steps to simplify.  The general procedure, however, was the same, i.e. take the log of both sides and then solve.