Solved example of logarithmic equations
Grouping terms
The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$
Rewrite the number $3$ as a logarithm of base $10$
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 $b$
Multiply both sides of the equation by $x-1$
Solve the product $1000\left(x-1\right)$
We need to isolate the dependent variable $x$, we can do that by subtracting $1$ from both sides of the equation
Grouping terms
Divide both sides of the equation by $-999$
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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