Solved example of logarithmic equations
Apply the formula: $a\log_{b}\left(x\right)$$=\log_{b}\left(x^a\right)$
The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$
Take the variable outside of the logarithm
Any expression (except $0$ and $\infty$) to the power of $0$ is equal to $1$
Simplifying the logarithm
Multiply both sides of the equation by $x+6$
Move everything to the left hand side of the equation
Factor the trinomial $x^2-x-6$ finding two numbers that multiply to form $-6$ and added form $-1$
Thus
Break the equation in $2$ factors and set each equal to zero, to obtain
Solve the equation ($1$)
We need to isolate the dependent variable , we can do that by simultaneously subtracting $2$ from both sides of the equation
Canceling terms on both sides
$x+0=x$, where $x$ is any expression
Canceling terms on both sides
Solve the equation ($2$)
We need to isolate the dependent variable , we can do that by simultaneously subtracting $-3$ from both sides of the equation
Canceling terms on both sides
$x+0=x$, where $x$ is any expression
Canceling terms on both sides
Combining all solutions, the $2$ solutions of the equation are
Verify that the solutions obtained are valid in the initial equation
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
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