Solved example of logarithmic equations
Express the numbers in the equation as logarithms of base $10$
Grouping terms
The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$
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
Eliminate the $-999$ from the left side, multiplying both sides of the equation by the inverse of $-999$
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
Access detailed step by step solutions to thousands of problems, growing every day!