Question 819953: solve for x and check your answer. Justify each step in the solution process.
Log5(3xsquared-1)= Log5(2x) Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
Solving problems like this usually starts with using algebra and/or properties of logarithms to transform the equation into one of the following general forms:
log(expression) = number
or
log(expression) = log(other_expression)
(The two logs in the second form must have the same base.)
But your equation is already in the second form! So we can move on to the next step. The next step with the second form is based on some simple logic. The only way logarithms (with the same base) of two expressions can be equal is if the two expressions themselves are equal, too. So:
Next we move on to solving the equation (now that the variables are out of logarithms). This equation is a quadratic equation so we want one side to be zero. Subtracting 2x from each side we get:
Now we factor:
(3x+1)(x-1) = 0
Next the Zero Product Property:
3x+1 = 0 or x-1 = 0
Solving these we should get: or
Next we check. This is not optional! A check must be made to ensure that all bases and arguments of all logarithms are valid. (Valid bases are positive but not 1 and valid arguments are positive). If a "solution" makes any base or argument of any logarithm invalid then it must be rejected!
Use the original equation to check:
Checking :
Simplifying...
Already we can see a problem. The argument on the right side is negative (i.e. invalid). So we must reject this "solution".
Checking :
Simplifying...
Check! Both bases are 5's (valid) and both arguments are 2's (also valid).
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