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Welcome back bilgate — missed you this week. Here are the solutions for the questions from the image: 41. Calculate the value of x if 2^x = 16. Step 1: Express 16 as a power of 2. 16 = 2^4 Step 2: Equate the exponents. 2^x = 2^4 x = 4 The value of x is 4. 42. Calculate the value of y if 3^y = (1)/(27). Step 1: Express (1)/(27) as a power of 3. (1)/(27) = (1)/(3^3) = 3^-3 Step 2: Equate the exponents. 3^y = 3^-3 y = -3 The value of y is -3. 43. Calculate the value of z if 5^z = sqrt(125). Step 1: Express sqrt(125) as a power of 5. sqrt(125) = sqrt(5^3) = 5^(3)/(2) Step 2: Equate the exponents. 5^z = 5^(3)/(2) z = (3)/(2) The value of z is (3)/(2). 44. Calculate the value of p if 4^p = 8. Step 1: Express both 4 and 8 as powers of the same base (2). 4 = 2^2 8 = 2^3 Step 2: Substitute these into the equation and simplify. (2^2)^p = 2^3 2^2p = 2^3 Step 3: Equate the exponents. 2p = 3 p = (3)/(2) The value of p is (3)/(2). 45. Calculate the value of q if 27^q = 9. Step 1: Express both 27 and 9 as powers of the same base (3). 27 = 3^3 9 = 3^2 Step 2: Substitute these into the equation and simplify. (3^3)^q = 3^2 3^3q = 3^2 Step 3: Equate the exponents. 3q = 2 q = (2)/(3) The value of q is (2)/(3). 46. Calculate the value of r if 1000^r = 100. Step 1: Express both 1000 and 100 as powers of the same base (10). 1000 = 10^3 100 = 10^2 Step 2: Substitute these into the equation and simplify. (10^3)^r = 10^2 10^3r = 10^2 Step 3: Equate the exponents. 3r = 2 r = (2)/(3) The value of r is (2)/(3). 47. Calculate the value of s if 64^s = 4. Step 1: Express both 64 and 4 as powers of the same base (4). 64 = 4^3 4 = 4^1 Step 2: Substitute these into the equation and simplify. (4^3)^s = 4^1 4^3s = 4^1 Step 3: Equate the exponents. 3s = 1 s = (1)/(3) The value of s is (1)/(3). 48. Calculate the value of t if 81^t = 3. Step 1: Express both 81 and 3 as powers of the same base (3). 81 = 3^4 3 = 3^1 Step 2: Substitute these into the equation and simplify. (3^4)^t = 3^1 3^4t = 3^1 Step 3: Equate the exponents. 4t = 1 t = (1)/(4) The value of t is (1)/(4). 49. Calculate the value of u if 125^u = 5. Step 1: Express both 125 and 5 as powers of the same base (5). 125 = 5^3 5 = 5^1 Step 2: Substitute these into the equation and simplify. (5^3)^u = 5^1 5^3u = 5^1 Step 3: Equate the exponents. 3u = 1 u = (1)/(3) The value of u is (1)/(3). 50. Calculate the value of v if 16^v = 2. Step 1: Express both 16 and 2 as powers of the same base (2). 16 = 2^4 2 = 2^1 Step 2: Substitute these into the equation and simplify. (2^4)^v = 2^1 2^4v = 2^1 Step 3: Equate the exponents. 4v = 1 v = (1)/(4) The value of v is (1)/(4). Send me the next one 📸