To find the exact value of each logarithm, we need to determine the exponent that the base must be raised to in order to get the given number.
a. For log749\log _{7} 49log749, we need to find the exponent x x x such that 7x=49 7^x = 49 7x=49.
b. For log44\log _{4} 4log44, we need to find the exponent x x x such that 4x=4 4^x = 4 4x=4.
c. For log1/232\log _{1 / 2} 32log1/232, we need to find the exponent x x x such that (1/2)x=32 (1/2)^x = 32 (1/2)x=32.
To find log749\log _{7} 49log749, we need to determine the exponent x x x such that 7x=49 7^x = 49 7x=49.
Since 49=72 49 = 7^2 49=72, we have: 7x=72 7^x = 7^2 7x=72 Thus, x=2 x = 2 x=2.
To find log44\log _{4} 4log44, we need to determine the exponent x x x such that 4x=4 4^x = 4 4x=4.
Since 4=41 4 = 4^1 4=41, we have: 4x=41 4^x = 4^1 4x=41 Thus, x=1 x = 1 x=1.
To find log1/232\log _{1 / 2} 32log1/232, we need to determine the exponent x x x such that (12)x=32 \left(\frac{1}{2}\right)^x = 32 (21)x=32.
Since 32=25 32 = 2^5 32=25, we can rewrite the equation as: (12)x=25 \left(\frac{1}{2}\right)^x = 2^5 (21)x=25 This implies: 2−x=25 2^{-x} = 2^5 2−x=25 Thus, −x=5 -x = 5 −x=5 and x=−5 x = -5 x=−5.
x=2,1,−5\boxed{x = 2, 1, -5}x=2,1,−5
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