Use logarithmic properties and the fact that ln(2) ≈ 0.69 and ln(3) ≈ 1.10 to approximate the value of each of the
following logarithmic expressions. do not use a calculator.
ln(108)

ln(108) = 3.58

Step-by-step explanation:

Data provided:

ln(2) ≈ 0.69

and

ln(3) ≈ 1.10

equation to be solved

ln(108)

now,

ln(108) = ln (2 × 54)

or

= ln( 2 × 2 × 27 )

or

= ln( 2 × 2 × 3 × 3 × 3 )

or

= ln( 2² × 3³ )

Now,

we have the property of log function as:

1) ln(AB) = ln(A) + ln(B)

and,

2) ln(xⁿ) = n × ln(x)

on applying the above properties , we get

ln( 2² × 3³ ) = ln(2²) + ln(3³)        using property (1)

or

= 2ln(2) + 3ln(3)                  using property (2)

on substituting the values given in the question

we get

ln(108) = ( 2 × 0.69 ) + ( 2 × 1.10 )

= 1.38 + 2.20

= 3.58

x ≥ 18 is your answer

step-by-step explanation:

let "a number" = x

(2/3)x + 17 ≥ 29

isolate the variable (x). do the opposite of pemdas (parenthesis, exponent (& root), multiplication, division, addition, subtraction).

first, subtract 17 from both sides.

(2/3)x + 17 (-17) ≥ 29 (-17)

(2/3)x ≥ 29 - 17

(2/3)x ≥ 12

next, multiply 3/2 to both sides to isolate the x

(3/2)(2/3)x ≥ 12(3/2)

x ≥ 12(3/2)

simplify. multiply, then divide

x ≥ (12 * 3)/2

x ≥ (36)/2

x ≥ 18

x ≥ 18 is your answer

~

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is