Question

Solve all 3 Questions. 50 Points + Brainelist ​

Answer 1

Answer:

i)  Using log law:  $\log_aa=1$

$\implies \log_55+1=1+1=2$

ii) $\log \left(\dfrac{15}{8}\right)+4 \log 2-\log 3$

Using log law $a \log b=\log b^a$:

$\implies \log \left(\dfrac{15}{8}\right)+\log 2^4-\log 3$

$\implies \log \left(\dfrac{15}{8}\right)+\log 16-\log 3$

Using log law $\log a-\log b=\log (\frac{a}{b})$:

$\implies \log \left(\dfrac{15}{8}\right)+\log\left(\dfrac{16}{3}\right)$

Using log law  $\log a+\log b=\log(ab)$:

$\implies \log \left(\dfrac{15}{8}\cdot \dfrac{16}{3}\right)$

$\implies \log 10$

Using log law:  $\log_aa=1$

$\implies \log_{10} 10=1$

iii) Take log of base 10:

$\log_{10}(\sqrt{8.357}\times0.895^2)$

$\implies \dfrac12\log_{10}(8.357)+2\log_{10}(0.895)$

Log tables

The characteristic of the logarithm of a number is the exponent of 10 in its scientific notation.

The mantissa is found using the log tables and is always prefixed by a decimal point.

The row is the first two non-zero digits of the number, and the column is the 3rd digit of the number

Use the log tables to find $\log_{10}(8.357)$:

8.357 = 8.357 × 10⁰

⇒ characteristic = 0

log table:  row 83, column 5 ⇒ mantissa 9217

(as there is a 4th digit) Mean difference 7 = 4

mantissa + mean difference = 9217 + 4 = 9221 ⇒ 0.9221

characteristic + mantissa = 0 + 0.9221 = 0.9221

Therefore, $\log_{10}(8.357)=0.9221$

Use the log tables to find $\log_{10}(0.895)$:

$0.895 = 8.95\times 10^{-1}$

⇒ characteristic = -1

log table: row 89, column 5 ⇒ mantissa 9518⇒ 0.9518

characteristic  + mantissa = -1 + 0.9518= -0.0482

Therefore, $\log_{10}(0.895)=-0.0482$

Therefore,

$\implies \dfrac12\log_{10}(8.357)+2\log_{10}(0.895)$

$\implies \dfrac12 \cdot 0.9221+2\cdot-0.0482$

$\implies 0.36465$

Therefore,

$\log_{10}(\sqrt{8.357}\times0.895^2)=0.36465$

Using $\log_ab=c \implies a^c=b$

$\implies \sqrt{8.357}\times0.895^2=10^{0.36465}$

$\implies \sqrt{8.357}\times0.895^2=2.3155$