SOLUTION: Please help solve this... log9(x-7)+log9(x-7)=1

Question 114927: Please help solve this...
log9(x-7)+log9(x-7)=1
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Given:
.

.
Since the base of the two logarithms on the left side is the same, the fact that they are
added means that by the rules of logarithms they can be combined to a single logarithm of
their products. In other words they combine as follows:
.

.
You can get the same result if you look at it this way ...
.
Recognize that involves two identical terms and so the sum
just equals 2 times one of the terms. In other words the sum is .
And by the rules of logarithms, the multiplier of a logarithm can be used as an exponent of
the quantity on which the logarithm is operating. So the multiplier 2 becomes the exponent
2 and the sum of the two logarithms equals which in turn equals

.
Back to the problem. We have now found an equivalent form of the left side of the equation
and therefore the equation becomes:
.

.
Now convert from this logarithmic form to the exponential form of this equation using the
conversion rule:
.
is equivalent to the exponential form
.
In our problem: a = 9, B = (x - 7)^2, and y = 1. Substituting these values into the exponential
form of the logarithmic equation results in our equation becoming:
.

.
Raising 9 to the exponent 1 just makes it 9 so the equation simplifies to:
.

.
Take the square root of both sides. The square root of 9 is 3 and the square root of
. Substituting this information into the equation reduces the equation to:
.

.
Solve for x by getting rid of the -7 on the right side. To do that just add +7 to both
sides and you have:
.

.
Hope this helps you see your way through this problem.
.
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