Evaluate In 14553 if In 3=1.0986, In 7=1.9459 and In 11=2.3979.

Kwanza, tunagawanya 14553 kwa nambari kuu ndogo zaidi.
$14553 \div 3 = 4851$
$4851 \div 3 = 1617$
$1617 \div 3 = 539$
Hivyo, $14553 = 3^3 \times 539$.

Sasa, tunagawanya 539:
$539 \div 7 = 77$
$77 \div 7 = 11$
Hivyo, $539 = 7^2 \times 11$.

Kwa hiyo, vigawo vikuu vya 14553 ni $3^3 \times 7^2 \times 11$.

Tunajua kuwa $\ln (ab) = \ln a + \ln b$ na $\ln (a^n) = n \ln a$.

$$\ln 14553 = \ln (3^3 \times 7^2 \times 11)$$ $$\ln 14553 = \ln (3^3) + \ln (7^2) + \ln 11$$ $$\ln 14553 = 3 \ln 3 + 2 \ln 7 + \ln 11$$

Tunapewa:
$\ln 3 = 1.0986$
$\ln 7 = 1.9459$
$\ln 11 = 2.3979$

Badilisha maadili haya kwenye mlinganyo:

$$\ln 14553 = 3(1.0986) + 2(1.9459) + 2.3979$$

$$3 \times 1.0986 = 3.2958$$ $$2 \times 1.9459 = 3.8918$$

Sasa jumlisha nambari hizi:

$$\ln 14553 = 3.2958 + 3.8918 + 2.3979$$ $$\ln 14553 = 7.1876 + 2.3979$$ $$\ln 14553 = 9.5855$$

Thamani ya $\ln 14553$ ni:

$$9.5855$$