```Question 601341
Since no other tutor has tried this one, I'll make an attempt.
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You are asked to solve for theta given that:
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{{{9^(cos(theta)) = 1/3}}}
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First, recognize that 9 is equal to 3 squared. So we can substitute 3-squared for 9 and get the equation:
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{{{(3^2)^(cos(theta)) = 1/3}}}
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Then by the power rule of exponents, you can multiply {{{2*cos(theta)}}} on the left side of this equation to make the equation become:
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{{{3^(2*cos(theta)) = 1/3}}}
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Since the unknown that we are trying to solve for is in an exponent, let's think in terms of taking the logarithm of both sides. If we take the logarithm of both sides (we can use base 10) we get:
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{{{log(3^(2*cos(theta))) = log(1/3)}}}
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By the rules of logarithms, on the left side we can bring the exponent out as a multiplier to get:
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{{{(2*cos(theta))log (3) = log(1/3)}}}
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Using another rule of logarithms, we can treat the right side as the logarithm of a division, and this is equivalent to the difference in the logs of the numerator and the denominator. So the equation then becomes:
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{{{(2*cos(theta))log (3) = log(1) - log(3)}}}
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But the logarithm of 1 is equal to zero. So the first term on the right side disappears because it equals zero. This reduces the equation to:
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{{{(2*cos(theta))log (3) =  - log(3)}}}
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Next notice that we have the common factor of log(3) on both sides. So if we divide both sides by log(3) the equation simplifies to:
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{{{2*cos(theta)= -1}}}
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Then divide both sides of the equation by 2 to get:
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{{{cos(theta) = -1/2}}}
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If we limit the answer to the range in which theta is between zero and 360 degrees, we can say that the cosine is negative in the second and third quadrants. And since arccosine theta = 1/2 results in the value of theta being 60 degrees (or pi/3 if you prefer radians) then in the second quadrant the value of theta will be 120 degrees ((2*pi)/3 radians) and in the third quadrant the value of theta will be 240 degrees ((4*pi)/3 radians).
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In summary, the two answers are theta = 120 degrees and theta = 240 degrees within the restricted range of zero to 360 degrees. You can expand that range by adding multiple values of -360 degrees and +360 degrees to these two answers.
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Hope this helps you to understand the problem and gives some insight into how you can solve it.
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