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To explain this through the telephone, we can describe the steps as follows:
- Notice that the two logarithms in the equation have the same base. So, express it as one logarithm using the product property.
- So, at the left side of the equation write log_3 and multiply the two arguments x and (x+2). And the right side remains the same.
`log_3(x*(x+2)) = 1`
- Since `x*(x+2` ) is equal to `x^2+2x` , the argument of the logarithm becomes `x^2+2x` .
`log_3(x^2+2x) = 1`
- Since the logarithm of `x^2+2x` is equal to 1, then we apply the property that a logarithm is equal to 1 if its base and argument are the same `(log_b b=1)` .
- This means that the argument of `log_3` is equal to 3. So, set `x^2+2x` equal to 3. And this becomes our new equation.
`x^2+2x = 3`
- Now that we have a quadratic equation, to solve for x, set one side equal to zero. To do this, subtract both sides by 3.
`x^2+2x - 3 = 3-3`
`x^2+2x - 3 =0`
Then, factor left side.
`(x+3)(x - 1) = 0`
Next, set each factor equal to zero.
`x+ 3= 0` and `x-1=0`
So the equation breaks into two.
For the first equation, subtract both sides by 3.
Note that in logarithm, we can not have a negative number as its argument. So, we do no consider -3 as a solution to our equation.
Next, solve for x in the second equation. To do so, add both sides by 1.
Hence, the solution to the given equation` log_3x + log_3(x+1)` is `x=1` .
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