Math Questions With Solutions (3): Logarithms and Exponentials

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
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) - Log2(100)
D. Log
8(4)
E. Log
5(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 + 3x + 3x-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 = 255/2
D. Log(x - 1) = Log(6) - Log(x)
E. Log
3(3x+1 - 18) = 2
F. Log
7(x) + Logx(7) = 2
G. Log
x(27) = 3/4

Solutions The Above Questions

Solution to Question 1
A. 1040.5 = 104 * 0.5 = 102 = 100
B. 10log(9) / 2 = 10log(91/2) = 3
C. e3 ln(4) = e ln(43) = 43 = 64
D. 10-2 Log(4) = 10Log(4-2) = 4-2 = 1/16
E. e2 Ln(4) - ln(16) = eLn(42) - ln(16) = e0 = 1

Solution to Question 2
A. Log9(3) = Log9(91/2) = 1/2
B. Log2(1/8) = Log2(2-3) = -3
C. Log2(400) - Log2(100) = Log2(400/100) = 2
D. Log8(4) = Log8(82/3) = 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
3x+2 + 3x+1 + 3x = 39 *3
Factor 3x out: 3x(9 + 3 + 1) = 39 * 3
Simplify : 3x = 9 and solve: x = 2

Solution to Question 4
A. Solve for ln(x): ln(x) = e4 , solve for x: x = ee4
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: 3x+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: x3/4 = 27 , solve for x: x = 34

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