SOLUTION: Ok so I have the solutions to the problem...but I keep coming up with the wrong answer and I don't understand why. Here's the problem: {{{16^(d-4)=3^(3-d)}}} The instructions

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Ok so I have the solutions to the problem...but I keep coming up with the wrong answer and I don't understand why. Here's the problem: {{{16^(d-4)=3^(3-d)}}} The instructions      Log On


   



Question 195522: Ok so I have the solutions to the problem...but I keep coming up with the wrong answer and I don't understand why. Here's the problem:
16%5E%28d-4%29=3%5E%283-d%29
The instructions are to solve each equation or inequality. Round to four decimal places.
Here's what I've tried:
16%5E%28d-4%29=3%5E%283-d%29
log%2816%5E%28d-4%29%29=log%283%5E%283-d%29%29
d%28log%2816%29-4log%2816%29%29=d%28log%283%29%2B3log%283%29%29
d=%28log%283%29%2B3log%283%29%29%2F%28log%2816%29-4log%2816%29%29
The book says that the solution is 3.716

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

You went wrong on the rule of logarithms that says

log%28X%5EY%29=Y%2Alog%28X%29

You move the entire exponent in front of the log:


16%5E%28d-4%29=3%5E%283-d%29

Take logs of both sides

log%2816%5E%28d-4%29%29=log%283%5E%283-d%29%29

Now use the rule that you missed.  Move the
(d-4) in parentheses in front of the log on
the left, and move the (3-d) in parentheses 
in front of the log on the right, and 
get:

%28d-4%29log%2816%29=%283-d%29log%283%29

Now here is something that might make it
easier for you, since it is easier to work
with single letters than logs.  So I suggest
doing this:

Let the letter A = log(16)
Let the letter B = log(3)

Substitute the letter A for log(16) and
substitute the letter B for log(3)

%28d-4%29A=%283-d%29B

A%28d-4%29=B%283-d%29

Ad-4A=3B-Bd

Ad%2BBd=3B%2B4A

d%28A%2BB%29=3B%2B4A

d=%283B%2B4A%29%2F%28A%2BB%29

Now get your calculator and find

A = log(16) = 1.204119983
B = log(3) = .4771212547

Substitute those:

d=%283%2A.4771212547%2B4%2A1.204119983%29%2F%281.204119983%2B.4771212547%29

d=3.716208927

Edwin