SOLUTION: ln10+lnx^2=10

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Question 541271: ln10+lnx^2=10
Found 2 solutions by josmiceli, bucky:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+ln%2810%29+%2B+ln%28x%5E2%29+=+10+
+ln%2810%2Ax%5E2%29+=+10+
On my calculator, I want to
solve +x+=+e%5E10+ because
+10+=+ln%28x%29+
---------------
+e%5E10+=+22026.466+
Now I can say
+ln%28+10x%5E2%29+=+ln%28+22026.466%29+
+10x%5E2+=+22026.466+
+x%5E2+=+2202.647+
+x+=+46.932+
and
+x+=+-46.932+
-------------
check answer:
+ln%2810%29+%2B+ln%2846.932%5E2%29+=+10+
+ln%2810%29+%2B+ln%28x%5E2%29+=+10+
+ln%2810%29+%2B+ln%282202.613%29+=+10+
+2.303+%2B+7.697+=+10+
+10.0004+=+10+
close enough

Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Recall that by the rules of logarithms, when you add the logarithms of two different quantities, it is the same as having the logarithm of the product of those two quantities. In other words we can write the left side of the equation in your problem as:
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ln%2810%29+%2B+ln%28x%5E2%29+=+ln%2810%2Ax%5E2%29
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So go to your original problem and replace it left side with ln%2810%2Ax%5E2%29 to get:
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ln%2810%2Ax%5E2%29+=+10
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Next recall that in an equation of this form, you can just raise the base of the logarithm (in this case the base is e) to the exponent on the right side of the equation and set it equal to the quantity that the log is operating on to get an equivalent equation. In other words, raise e to the 10th power and set it equal to 10%2Ax%5E2:
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e%5E10+=+10%2Ax%5E2
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swap sides (transpose) to get the unknown x on the left side and the constant on the right side (just to put into a little more conventional form):
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10%2Ax%5E2+=+e%5E10
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Divide by the coefficient of the x^2 term (that is divide both sides by 10):
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x%5E2+=+%28e%5E10%29%2F10
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Don't lose sight of the fact that the right side of this equation is just a number that you can compute with a scientific calculator. We can either do that now and then solve for x by taking the square root of both sides, or we can take the square root of both sides of the above equation now. Don't forget that the square root of a term with an exponent is found by dividing the exponent by 2. So taking the square root of both sides results in:
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sqrt%28x%5E2%29+=+sqrt%28e%5E10%2F10%29
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x+=+sqrt%28e%5E10%29%2Fsqrt%2810%29
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x+=+e%5E5%2Fsqrt%2810%29
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It is against convention to have a radical in the denominator so we can eliminate that by multiplying by sqrt%2810%29%2Fsqrt%2810%29 which is equivalent to multiplying the right side by 1 to get:
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x=%28e%5E5%2Asqrt%2810%29%29%2F%28sqrt%2810%29%2Asqrt%2810%29%29
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The multiplication in the denominator becomes just 10 and so the answer is:
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x+=+%28e%5E5%2Asqrt%2810%29%29%2F10
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And don't forget that the square root function can result in both positive and negative answers similar to what you get from the plus and minus radical in the quadratic formula. So the two answers to this problem are:
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x+=+%28e%5E5%2Asqrt%2810%29%29%2F10 or x+=+-%28%28e%5E5%2Asqrt%2810%29%29%2F10%29
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If you evaluate these two answers on a calculator you can get numerically equivalent answers (but not exact answers because of non-terminating decimal places) of:
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x=46.9323617505 or x=-46.9323617505
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Hope this helps you to understand this problem a little better and also helps to refresh your memory about a couple of the rules of logarithms.
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