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Solving equations where the variable is in the argument (or base) of a logarithm usually starts with using algebra and/or properties of logarithms to transform the equation into on of the following forms:
log(expression) = other_expression
or
log(expression) = log(other_expression)
Since your equation has a "non-log" term, the 2, it will be difficult to reach the "all-log" second form. So we will aim for the first form. First we will gather the log terms on one side. Subtracting from each side we get:
Now we need to find a way to express the two logs on the left as a single log. Fortunately one of the properties of logarithms, , gives us a way to do this:
We now have the first form.
The next step with the first form is to rewrite the equation in exponential form. In general is equivalent to . Using this pattern on our equation we get:
which simplifies to:
We now have an equation without logs and where the variable is "out in the open". Now we use algebra to solve for x. Let's start by getting rid of the fraction. Multiplying both sides by (x-4) we get:
25x - 100 = x - 2
Subtracting x from both sides:
24x - 100 = -2
Adding 100 to each side:
24x = 98
Dividing by 24 we get:
Checking solutions for equations like this one is not optional. You must ensure that all arguments to all logarithms are positive. Use the original equation to check:
Checking x = 12/49:
We can already see that not just one but both arguments will be negative is x = 12/49. So we must reject this solution. And since this was the only solution we found, there is no solution to your equation.
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