Math Questions With Solutions (3) Logarithms and Exponentials

A set of math questions related to exponential and logarithmic expressions and equations are presented. The solutions are provided and are located at the lower part of the page.

Question 1

Evaluate the given exponential expressions without using a calculator.
A. 10^{4^0.5}
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. Log₉(3)
B. Log₂(1/8)
C. Log₂(400) - Log₂(100)
D. Log₈(4)
E. Log₅(600) - Log₅(24)

Question 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

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. Log₃(3^x+1 - 18) = 2
F. Log₇(x) + Log_x(7) = 2
G. Log_x(27) = 3/4

Solutions The Above Questions

Solution to Question 1
A. 10^{4^0.5} = 10^{4 * 0.5} = 10² = 100
B. 10^{log(9) / 2} = 10^{log(9^1/2)} = 3
C. e^{3 ln(4)} = e^ln(4³) = 4³ = 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)} = e⁰ = 1

Solution to Question 2
A. Log₉(3) = Log₉(9^1/2) = 1/2
B. Log₂(1/8) = Log₂(2^-3) = -3
C. Log₂(400) - Log₂(100) = Log₂(400/100) = 2
D. Log₈(4) = Log₈(8^2/3) = 2/3
E. Log₅(600) - Log₅(24) = Log₅(600/24) = Log₅(25) = 2

Solution to Question 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

Solution to Question 4
A. Solve for ln(x): ln(x) = e⁴ , solve for x: x = e^e⁴
B. rewrite equation: ln(x/4) = ln(x²/16) , gives: x/4 = x²/16 , solve for x: x = 4
C. rewrite equation: (ln(x))⁵ = 5⁵ , gives: ln(x) = 5 , solve for x: x = e⁵
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 = 3², 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 = 3⁴

Math Questions With Solutions (3) Logarithms and Exponentials

Question 1

Question 2

Question 3

Question 4

Solutions The Above Questions

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