You can put this solution on YOUR website!
Solving equations like this usually starts with transforming the equation into one of the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)
If we could find a way to combine the two logs into one, we would have the first form. We cannot simply subtract them because they are not like terms. (Like logarithmic terms have the same bases and same arguments.) But there is a property of logarithms, , which gives us another way to combine two logs that have a "-" between them. This property requires that the bases be the same and the coefficients be 1's. Our logs meet both requirements. Using the property on your logs we get:
We now have the first form.
The next step with this form is to rewrite the equation in exponential form. In general is equivalent to . Using this pattern on our equation (and using the fact that the base of log is 10) we get:
which simplifies to:
Now that the logs are gone we can solve the equation. Multiplying both sides by x we get:
Subtracting x we get:
Dividing by 99:
When solving logarithmic equations like this you must check your solution(s). It is not optional! You must ensure that all arguments of all logs are positive for each solution. If a "solution" makes any argument of any log zero or negative we must reject that "solution".
Use the original equation to check:
Checking x = 2/99:
We can already see that both arguments, 2/99+2 and 2/99, will be positive if x = 2/99. So this solution passes the check! (If 2/99 had failed this check then we would reject it. And since it was the only "solution" we found rejecting it would mean that the equation has no solutions.)
So the only solution to your equation is x = 2/99.
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