** Final answer to the problem

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** Step-by-step Solution **

** How should I solve this problem?

- Choose an option
- Solve for x
- Find the derivative using the definition
- Solve by quadratic formula (general formula)
- Simplify
- Find the integral
- Find the derivative
- Factor
- Factor by completing the square
- Find the roots
- Load more...

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Apply the formula: $a\log_{b}\left(x\right)$$=\log_{b}\left(x^a\right)$

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The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$

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Rewrite the number $0$ as a logarithm of base $10$

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For two logarithms of the same base to be equal, their arguments must be equal. In other words, if $\log(a)=\log(b)$ then $a$ must equal $b$

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Multiply both sides of the equation by $x+6$

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Move everything to the left hand side of the equation

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Factor the trinomial $x^2-x-6$ finding two numbers that multiply to form $-6$ and added form $-1$

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Thus

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Break the equation in $2$ factors and set each equal to zero, to obtain

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Solve the equation ($1$)

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We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $2$ from both sides of the equation

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Canceling terms on both sides

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Solve the equation ($2$)

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We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-3$ from both sides of the equation

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Canceling terms on both sides

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Combining all solutions, the $2$ solutions of the equation are

Verify that the solutions obtained are valid in the initial equation

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The valid solutions to the logarithmic equation are the ones that, when replaced in the original equation, don't result in any logarithm of negative numbers or zero, since in those cases the logarithm does not exist

** Final answer to the problem

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