Solve each of the following equations without using calculator.
Problem 1 :
5^{x} = 625
Solution :
5^{x} = 625
625 is a power of 5 and it can be written as 5^{4}.
5^{x} = 5^{4}
Since the base is same on both sides, exponents can be equated.
x = 4
Problem 2 :
Solution :
Problem 3 :
Solution :
Problem 4 :
3^{x  10} = 27^{x}
Solution :
3^{x  10} = 27^{x}
3^{x  10} = (3^{3})^{x}
3^{x  10} = 3^{3}^{x}
x  10 = 3x
2x = 10
x = 5
Problem 5 :
4^{x + 2} = 8^{x}
Solution :
4^{x + 2} = 8^{x}
(2^{2})^{x + 2} = (2^{3})^{x}
2^{2(}^{x + 2)} = 2^{3}^{x}
2(x + 2) = 3x
2x + 4 = 3x
4 = x
Problem 6 :
Solution :
x^{2} + 1 = 2x
x^{2}  2x + 1 = 0
Solve by factoring.
x^{2}  x  x + 1 = 0
x(x  1)  1(x  1) = 0
(x  1)(x  1) = 0
(x  1)^{2} = 0
Taking square root on both sides,
x  1 = 0
x = 1
Problem 7 :
Solution :
2(x + 4) = 3x^{2}
3x^{2 } 2x  8 = 0
Solve by factoring.
3x^{2 } 6x + 4x  8 = 0
3x(x^{ } 2) + 4(x  2) = 0
(x  2)(3x + 4) = 0
x  2 = 0 or 3x + 4 = 0
x  2 = 0 x = 2 

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