Question 131921
Given to solve for x:
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{{{100^(x^2)=10^(5x-3)}}}
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Take the base 10 logarithm of both sides:
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{{{log(10,100^(x^2)) = log(10,10^(5x-3))}}}
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Apply the rule of logarithms that says an exponent comes out as the multiplier of the logarithm.
When you apply that rule, the problem becomes:
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{{{(x^2)*log(10,100) = (5x-3)log(10,10)}}}
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But your calculator will tell you that {{{log(10,100) = 2}}} and {{{log(10,10) = 1}}}
Make these two substitutions into the equation and it becomes:
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{{{(x^2)*2 = (5x - 3)*1}}}
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which simplifies to:
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{{{2x^2 = 5x - 3}}}
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subtract 5x - 3 from both sides to get the equation into the standard form of:
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{{{2x^2 - 5x + 3 = 0}}}
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The left side of this equation factors to give:
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{{{(x -1)*(2x-3) = 0}}}
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Notice that this equation will be true if either of the two factors is equal to zero ... because
a multiplication by zero on the left side makes the entire left side equal to zero and 
therefore equal to the right side.
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So either {{{x-1 = 0}}} or {{{2x - 3= 0}}}
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Solve these two equations. From {{{x-1 = 0}}} you can see that {{{x = 1}}} and from {{{2x-3= 0}}}
you get {{{x = 3/2 = 1.5}}}
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Therefore, the two answers to this problem are x = 1 and x = 1.5
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Hope this helps you to understand the problem and how it can be worked to get the answer.
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