Practice | Hashtable

41. If the logarithm of a number is - 3.153, what are characteristic and mantissa?

A. characteristic = -4, mantissa = 0.847

B. characteristic = -3, mantissa = -0.153

C. characteristic = 4, mantissa = -0.847

D. characteristic = 3, mantissa = -0.153

Solution:

\begin{aligned} let \log x = - 3.153 \\ then, \log x = - 3.153 \\ = - 3 + (- 0.153) \\ = (- 3 - 1) + (1 - 0.153) \\ = - 4 + 0.847 = \bar{4} .847 \\ Hence, \\ characteristic  =   - 4, mantissa  =  0 .847 \end{aligned}

42. If $\log 2 = 0.30103,$ the number of digits in $4^{50}$ is -

A. 30

B. 31

C. 100

D. 200

Discuss

Solution:

\begin{aligned} \log 4^{50} \\ = 50 \log 4 \\ = 50 \log 2^{2} \\ = (50 \times 2) \log 2 \\ = 100 \times \log 2 \\ = (100 \times 0.30103) \\ = 30.103 \\ ∴ characteristic = 30, \end{aligned}

Hence, the number of digits in

4^{50} = 31

43. The number of digits in $4^{9} \times 5^{17},$ when expressed in usual form, is -

A. 16

B. 17

C. 18

D. 19

Discuss

Solution:

\begin{aligned} \log (4^{9} \times 5^{17}) \\ = \log (4^{9}) + \log (5^{17}) \\ = \log {(2^{2})}^{9} + \log (5^{17}) \\ = \log (2^{18}) + \log (5^{17}) \\ = 18 \log 2 + 17 \log 5 \\ = 18 \log 2 + 17 (\log 10 - \log 2) \\ = 18 \log 2 + 17 \log 10 - 17 \log 2 \\ = \log 2 + 17 \log 10 \\ = 0.3010 + 17 \times 1 = 17.3010 \\ ∴ Characteristic = 17 \\ Hence, the number of digits in (4^{9} \times 5^{17}) = 18 \end{aligned}

44. If $\log 3 \log (3^{x} - 2)$ and $\log (3^{x} + 4)$ are in arithmetic progression, then x is equal to

A. $\frac{8}{3}$

B. $\log 3^{8}$

C. $\log 2^{3}$

D. 8

Discuss

Solution:

In arithmetic progression common ratio are equal to

\log (3^{x} - 2) - \log 3 =

\log (3^{x} + 4) -

\log (3^{x} - 2)

\frac{\log (3^{x} - 2)}{\log 3} = \frac{\log (3^{x} + 4)}{\log (3^{x} - 2)}

(∴ \log a - \log b = \log \frac{a}{b})

\begin{aligned} \frac{\log 3^{x}}{\log 2 \log 3} = \frac{x \log 3 \log 4 \log 2}{x \log 3} \\ \frac{x \log 3}{\log 2 \log 3} = \frac{x \log 3 \log 4 \log 2}{x \log 3} \\ \frac{x}{\log 2} = \log 4 \log 2 \\ x = \log 4 \log 2 \log 2 \\ x = \log 8 \\ x = \log 2^{3} \end{aligned}

45. If $\log_{10} a = p,$ $\log_{10} b = q,$ then what is $\log_{10} (a^{p} b^{q})$ equal to?

A. p² + q²

B. p² - q²

C. p²q²

D. $\frac{p^{2}}{q^{2}}$

Discuss

Solution:

\begin{aligned} Given, \\ \log_{10} a = p, \log_{10} b = q \\ \log_{10} (a^{p} b^{q}) = \log_{10} a^{p} + \log_{10} b^{q} \\ = p \log_{10} a + q \log_{10} b \\ = p^{2} + q^{2} \end{aligned}