SOLUTION: Please could someone help with the following with workings out please: 16^2-n = (1/4)^n+1. sixteen to the power two negative n equals a quarter to the power of n plus one.

Click here to see ALL problems on logarithm

Question 838174: Please could someone help with the following with workings out please:
16^2-n = (1/4)^n+1.
sixteen to the power two negative n equals a quarter to the power of n plus one.

In my assignment I can work out the ones like 128^n = 8 and (1/4)^n = 8, but I'm getting stuck with the more complex ones. Please could you explain how you get to each stage if possible. Many thanks in advance, David.

Found 2 solutions by jsmallt9, Theo:
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
First of all, please put exponents which are not just a variable or single number in parentheses. If you have not also explained your expression in English I would not have been sure what your equation was.

Equations with variables in the exponents can usually be solved using logarithms. But if it is possible to rewrite the equation so that each side is a power of the same base, then there is an easier and more accurate way than using logarithms. 16 and 1/4 are powers of each other but what powers these are may not be obvious. But they are both powers of 2 (which should be more obvious). So the solution will be easier if we start by rewriting each side as powers of 2:

For powers of a power the rule is multiply the exponents:

The next step is based on simple logic. The only way two powers of 2 can be equal, as the now equation says they are, is if the exponents are equal, too. So:
8-4n = -2n-2
Now we solve this simple equation. Adding 4n to each side:
8 = 2n-2
Adding 2:
10 = 2n
Dividing by 2:
5 = n

P.S. If we had used logarithms we might not have gotten an answer of 5. Because of the rounded decimals which a calculator uses, we might have gotten something like 5.00001 or 4.99998. So the method we used above should be used whenever possible.

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
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.