Question 1
Evaluate the given exponential expressions without using a calculator.
A. 10^{40.5}
B. 10^{log(9) / 2}
C. e^{3 ln(4)}
D. 10^{2 Log(4)}
E. e^{2 Ln(4)  ln(16)}
Question 2
Evaluate the following logarithmic expressions without using a calculator.
A. Log_{9}(3)
B. Log_{2}(1/8)
C. Log_{2}(400)  Log_{2}(100)
D. Log_{8}(4)
E. Log_{5}(600)  Log_{5}(24)
Question 3
Solve the following exponential equations.
A. e^{x} = 4
B. e^{x} = ? ^{2}
C. 4^{log4(3x  2)} = 10
D. 10^{log10(3x)} = 15
E. 3^{x+1} + 3^{x} + 3^{x1} = 39
Question 4
Solve the following logaritmic equations.
A. ln(ln(x)) = 4
B. ln(x)  ln(4) = 2 ln(x)  ln(16)
C. (ln(x))^{5} = 25^{5/2}
D. Log(x  1) = Log(6)  Log(x)
E. Log_{3}(3^{x+1}  18) = 2
F. Log_{7}(x) + Log_{x}(7) = 2
G. Log_{x}(27) = 3/4
Solutions The Above Questions
Solution to Question 1
A. 10^{40.5} = 10^{4 * 0.5} = 10^{2} = 100
B. 10^{log(9) / 2} = 10^{log(91/2)} = 3
C. e^{3 ln(4)} = e^{ ln(43)} = 4^{3} = 64
D. 10^{2 Log(4)} = 10^{Log(42)} = 4^{2} = 1/16
E. e^{2 Ln(4)  ln(16)} = e^{Ln(42)  ln(16)} = e^{0} = 1
Solution to Question 2
A. Log_{9}(3) = Log_{9}(9^{1/2}) = 1/2
B. Log_{2}(1/8) = Log_{2}(2^{3}) = 3
C. Log_{2}(400)  Log_{2}(100) = Log_{2}(400/100) = 2
D. Log_{8}(4) = Log_{8}(8^{2/3}) = 2/3
E. Log_{5}(600)  Log_{5}(24) = Log_{5}(600/24) = Log_{5}(25) = 2
Solution to Question 3
A. x = ln(4)
B. x = 2 ln(?)
C. 3x  2 = 10 , x = 4
D. 3x = 15 , x = 5
E. Multiply all terms of the equation by 3 to obtain
3^{x+2} + 3^{x+1} + 3^{x} = 39 *3
Factor 3^{x} out: 3^{x}(9 + 3 + 1) = 39 * 3
Simplify : 3^{x} = 9 and solve: x = 2
Solution to Question 4
A. Solve for ln(x): ln(x) = e^{4} , solve for x: x = e^{e4}
B. rewrite equation: ln(x/4) = ln(x^{2}/16) , gives: x/4 = x^{2}/16 , solve for x: x = 4
C. rewrite equation: (ln(x))^{5} = 5^{5} , gives: ln(x) = 5 , solve for x: x = e^{5}
D. Rewrite as: Log(x  1) = Log(6/x) ,
gives: x  1 = 6/x, solve for x and check: x = 3 is the only solution.
E. Rewrite without Log: 3^{x+1}  18 = 3^{2}, solve for x: x = 2
F. Use change of base formula to rewrite equation:
ln(7) / ln(x) + ln(x) / ln(7) = 2
rewrite as: (ln(7))^{2} + (ln(x))^{2}  2ln(7) ln(x) = 0
( ln(7)  ln(x) )^{2} = 0 , which gives: ln(7) = ln(x)
solve for x: x = 7
G. Rewrite without Log: x^{3/4} = 27 , solve for x: x = 3^{4}
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