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)

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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(20056)   (Show Source): You can put this solution on YOUR website!



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
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!


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.)

So the only solution to your equation is x = 3.
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