__Questions 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)}

__Questions 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)

__Questions 3:__

Solve the following exponential equations.

A. e^{x} = 4

B. e^{x} = π ^{2}

C. 4^{log4(3x - 2)} = 10

D. 10^{log10(3x)} = 15

E. 3^{x+1} + 3^{x} + 3^{x-1} = 39

__Questions 4:__

Solve the following logarithmic 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

**ANSWERS TO ABOVE QUESTIONS**

__Answers to Questions 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(4-2)} = 4^{-2} = 1/16

E. e^{2 Ln(4) - ln(16)} = e^{Ln(42) - ln(16)} = e^{0} = 1

__Answers to Questions 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

__Answers to Questions 3:__

A. x = ln(4)

B. x = 2 ln(π)

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

__Answers to Questions 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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