SOLUTION: I need help with this equation... Can you please help me?? & can you please do it step by step??? thank you!!
log (n+2) + log 8 = log (n^2 + 7n + 10)
Question 251005: I need help with this equation... Can you please help me?? & can you please do it step by step??? thank you!!
log (n+2) + log 8 = log (n^2 + 7n + 10) Found 2 solutions by Edwin McCravy, jsmallt9:Answer by Edwin McCravy(20055) (Show Source):
Factor the right side:
Use this principle to rewrite the right side:
Subtract from both sides.
Use the principle: is equivalent to
to remove the single logs in front of both sides of the equation:
However we must check it in the ORIGINAL equation to
make sure it is a solution:
Use this principle to rewrite the left side:
It checks.
Edwin
Solving equations where the variable is in the argument of one or more logarithms usually involves transforming the equation into one fo the following forms:
log(expression) = other-expression
or
log(expression) = log(other-expression)
Since your equation has logarithms on both sides of the equation already, we will aim for the second form. We just have to find a way to combine the two logarithms on the left into a single logarithm. Fortunately we have a property of logarithms, , which allows us to do exactly what we want. Using this property on your equation gives us:
which simplifies to:
We now have the desired (second) form. The next step uses the idea that if the logarithm of n+2 is the same as the logarithm of then
n+2 must be the same as :
The variable is now out of the argument of any logarithms. This is why we use the desired forms. Equations in those forms can be rewritten without logarithms. (BTW: If you use the first form, you rewrite it in exponential form.)
We now have a "normal" quadratic equation to solve. So we'll get one side equal to zero (by subtracting 8n and 16 from ecah side):
Next we factor or use the Quadratic Formula. This factors pretty easily:
By the Zero Product Property this product can be zero only if one of the factors is zero. So: or
Solving these we get: or
With logarithmic equations it is important to check your answers. We must reject any solutions which make an argument of a logarithm negative or zero. Always use the original equation to check your answers.
Checking x = 3:
As you can see, all the arguments of the logarithms are positive. So we have no reason to reject x = 3. To finish the check we can use the property from before to combine the logarithms on the left side: Check!
Checking x = -2:
Already we have a problem. We have an argument to a logarithm that is zero. So we must reject this solution. (We only have to find one argument to a logarithm that is zero or negative to reject a solution. It makes no difference that the logarithm on the right ends up with a zero argument. Nor does it make any difference that log(8) is OK.)
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