Questions: Find the exact value of each logarithm without a calculator. a. log base 7 of 49 b. log base 4 of 4 c. log base 1/2 of 32

Find the exact value of each logarithm without a calculator.
a. log base 7 of 49
b. log base 4 of 4
c. log base 1/2 of 32
Transcript text: 8. Find the exact value of each logarithm without a calculator. a. $\log _{7} 49$ b. $\log _{4} 4$ c. $\log _{1 / 2} 32$
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Solution

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Solution Steps

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} 49, we need to find the exponent x x such that 7x=49 7^x = 49 .

b. For log44\log _{4} 4, we need to find the exponent x x such that 4x=4 4^x = 4 .

c. For log1/232\log _{1 / 2} 32, we need to find the exponent x x such that (1/2)x=32 (1/2)^x = 32 .

Step 1: Solve log749\log _{7} 49

To find log749\log _{7} 49, we need to determine the exponent x x such that 7x=49 7^x = 49 .

Since 49=72 49 = 7^2 , we have: 7x=72 7^x = 7^2 Thus, x=2 x = 2 .

Step 2: Solve log44\log _{4} 4

To find log44\log _{4} 4, we need to determine the exponent x x such that 4x=4 4^x = 4 .

Since 4=41 4 = 4^1 , we have: 4x=41 4^x = 4^1 Thus, x=1 x = 1 .

Step 3: Solve log1/232\log _{1 / 2} 32

To find log1/232\log _{1 / 2} 32, we need to determine the exponent x x such that (12)x=32 \left(\frac{1}{2}\right)^x = 32 .

Since 32=25 32 = 2^5 , we can rewrite the equation as: (12)x=25 \left(\frac{1}{2}\right)^x = 2^5 This implies: 2x=25 2^{-x} = 2^5 Thus, x=5 -x = 5 and x=5 x = -5 .

Final Answer

x=2,1,5\boxed{x = 2, 1, -5}

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