Using properties of logarithms in exercise, use the properties of logarithms and a fact that in 2≈ 0.6931 and in 3 ≈1.0986 to approximate the logarithm. then use a calculator to confirm your approximation.
(a) in 0.25
(b) in 24
(c) in (12)1/3
(d) in 1/72

Step-by-step explanation:

Given that ln 2 = 0.6931 and ln 3 = 1.0986

We have to find the values for the given numbers

using properties of log

Properties of log are

log a+log b = log ab:  log a-log b = log a/b

a) ln 0.25 = ln 1-ln 4 = -2ln 2 = -1.3862

b) ln 24 = ln 3 (2^3) = 3 ln 2 + ln 3 = 2.0793+1.0986

=3.1779

c) ln (12)^(1/3) = 1/3 ln (2^2*3)  =1/3 (1.3862+1.0986)

= 0.8283

d) ln (1/72) = -ln 72 = -ln 9 -ln 8

= -3 ln 3 - 3ln 2

= -5.2752

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 =

x = 3, y = 216 =

x = 4, y = 1296 =

x = 5, y = 7776 =

so, the geometric series obtained is .

thus, the common ratio is given by = = 6.

hence, the common ratio is 6.