SOLUTION: 4. Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. log(x+16)=log x+log 16

Question 825624: 4. Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer.
log(x+16)=log x+log 16

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
log(x+16) = log(x) + log(16)
First we use a property of logarithms, , to combine the two logs on the right into one:
log(x+16) = log(x*16)
or
log(x+16) = log(16x)

This equation says that two base 10 logs are equal. The only way this can be true is if the arguments are equal, too. So:
x+16 = 16x
Now we can solve for x. Subtracting x from each side:
16 = 15x
Dividing by 15:

One way to see if this result is in the domain is to try this solution in the original equation and see if all the arguments are valid (i.e. positive):

We should already be able to see that all three arguments will be positive. So 16/15 is in the domain and is a valid solution to the problem.
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