Solve for x: lg9 + 1 = lg(x+1) + lgx.

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log 9 + 1 = log(x+1) + log x

We know that log 10 = 1

==> log 9 + log 10 = log (x+1) + log x

Also, we know that: log a + log b = log a*b

==> log 9*10 = log (x+1)*x

==> log 90 =...

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log 9 + 1 = log(x+1) + log x

We know that log 10 = 1

==> log 9 + log 10 = log (x+1) + log x

Also, we know that: log a + log b = log a*b

==> log 9*10 = log (x+1)*x

==> log 90 = log (x^2 + x)

==> 90 = x^2 + x

==> x^2 + x - 90 = 0

Now let us factor:

==> (x+ 10 ) (x-9) = 0

==> x1= 9

==> x2= -10 (impossible because log x is not defined for negative values.

Then the answer is:

x = 9

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