7us logarithm tables to evaluate sqrt(0.0452×2.63/1.56)

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Haya, tutasuluhisha swali hili tena kwa undani zaidi.

Swali la 7: Tumia jedwali la logariti kutathmini $\sqrt{\frac{0.0142 \times 2.63}{1.56}}$ .

Wacha $X = \sqrt{\frac{0.0142 \times 2.63}{1.56}}$ .
Ili kutatua hili kwa kutumia logariti, tunachukua logariti ya pande zote mbili:

\log X = \log \left( \left(\frac{0.0142 \times 2.63}{1.56}\right)^{\frac{1}{2}} \right)

Tunatumia sheria za logariti: $\log(a^b) = b \log a$ , $\log(ab) = \log a + \log b$ , na $\log\left(\frac{a}{b}\right) = \log a - \log b$ .

\log X = \frac{1}{2} \left( \log(0.0142) + \log(2.63) - \log(1.56) \right)

Tafuta logariti za kila namba kutoka jedwali la logariti.
- $\log(0.0142)$ : Namba hii ina sifuri mbili kabla ya namba muhimu ya kwanza, hivyo tabia (characteristic) yake ni $\bar{2}$ . Mantisa ya 142 ni 0.1523. Hivyo, $\log(0.0142) = \bar{2}.1523$ .
- $\log(2.63)$ : Namba hii ina tarakimu moja kabla ya desimali, hivyo tabia yake ni 0. Mantisa ya 263 ni 0.4200. Hivyo, $\log(2.63) = 0.4200$ .
- $\log(1.56)$ : Namba hii ina tarakimu moja kabla ya desimali, hivyo tabia yake ni 0. Mantisa ya 156 ni 0.1931. Hivyo, $\log(1.56) = 0.1931$ .
Badilisha thamani hizi kwenye fomula ya $\log X$ na ufanye hesabu za kujumlisha na kutoa.
$\log X = \frac{1}{2} (\bar{2}.1523 + 0.4200 - 0.1931)$
Kwanza, jumlisha logariti za namba kwenye nambari:
$\bar{2}.1523 + 0.4200 = (-2 + 0.1523) + 0.4200 = -2 + (0.1523 + 0.4200) = -2 + 0.5723 = \bar{1}.5723$
Sasa, toa logariti ya namba kwenye kigawanyo:
$\bar{1}.5723 - 0.1931 = (-1 + 0.5723) - 0.1931 = -1 + (0.5723 - 0.1931) = -1 + 0.3792 = \bar{1}.3792$
Hivyo, $\log\left(\frac{0.0142 \times 2.63}{1.56}\right) = \bar{1}.3792$ .
Gawanya matokeo kwa 2.
$\log X = \frac{1}{2} (\bar{1}.3792)$
Ili kugawanya $\bar{1}.3792$ kwa 2, tunabadilisha tabia $\bar{1}$ ili iwe namba hasi inayoweza kugawanywa na 2 bila salio, huku tukihakikisha mantisa inabaki chanya.
$\bar{1}.3792 = -1 + 0.3792$ . Tunaweza kuandika $-1$ kama $-2 + 1$ .
Hivyo, $\bar{1}.3792 = -2 + 1 + 0.3792 = -2 + 1.3792$ .
Sasa gawanya kwa 2:
$\log X = \frac{1}{2} (-2 + 1.3792) = \frac{-2}{2} + \frac{1.3792}{2} = -1 + 0.6896 = \bar{1}.6896$
Tafuta antilog ya $\bar{1}.6896$ .
Mantisa 0.6896 inalingana na takriban 4.893 kutoka jedwali la antilogariti.
Tabia $\bar{1}$ inamaanisha kuzidisha kwa $10^{-1}$ .
$X = 4.893 \times 10^{-1}$ $X = 0.4893$
Jibu: 0.4893

7us logarithm tables to evaluate sqrt(0.0452x2.63/1.56)

Answer

Step-by-step solution

Tafuta logariti za kila namba kutoka jedwali la logariti.

Badilisha thamani hizi kwenye fomula ya $\log X$ na ufanye hesabu za kujumlisha na kutoa.

Gawanya matokeo kwa 2.

Tafuta antilog ya $\bar{1}.6896$ .

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