Logarithms of Numbers Less Than One (Worked Examples)

Lesson Progress

0% Complete

Topic Content:

Logarithms of Numbers Less Than One (Worked Examples)

Example 1.2.1:

Use logarithms to evaluate the following:

i) \( \scriptsize \left ( 0.624 \div 0.09835 \right )^2 \)

ii) \( \scriptsize\sqrt {0.6821 \times 0.005924} \)

iii) \( \sqrt [3] {\frac {65.91 \: \times \: 0.0741}{576.2}} \)

Solution:

i) \( \scriptsize \left ( 0.624 \div 0.09835 \right )^2 \)

Number		Logarithm
0.624		\(\scriptsize\bar{1}.7952\)
0.09835	–	\(\scriptsize\bar{2}.9927\)
		\(\scriptsize {0.8025 \: \times \: 2}\)
		\(\scriptsize 1.6050 \)
		Antilog = 4.0272
		Answer = \(\scriptsize 4.0272 \: \times \: 10^1 \\ \scriptsize = 40.272\)

ii) \( \scriptsize\sqrt {0.6821 \times 0.005924} \)

Number		Logarithm
0.6821		\(\scriptsize\bar{1}.8338 \)
0.005924	+	\(\scriptsize\bar{3}.7726\)
		\(\scriptsize\bar{3}.6064 \div 2\)
		\(\scriptsize(\bar{4} \: + \: 1.6064) \div 2\)
		\(\scriptsize\bar{2}.8032\)
		Antilog = 6.3562
		Answer = \( \scriptsize 6.356 \times 10^{-2}\\ \scriptsize \: or \: 0.06356 \)

iii) \( \sqrt [3] {\frac {65.91 \: \times \: 0.0741}{576.2}} \)

Number		Logarithm
65.91		\(\scriptsize 1.8190 \)
0.0741	+	\(\scriptsize\bar{2}.8698\)
		\(\scriptsize 0.6888 \)
567.2	–	\(\scriptsize 2.7537\)
		\(\scriptsize\bar{3}.9351 \: \div \: 3\)
		\(\scriptsize (\bar{3} \: + \: 0.9351) \: \div \: 3\)
		\(\scriptsize \bar{1}.3117\)
		Antilog = 2.0497
		Answer = \( \scriptsize 2.0497 \times 10^{-1}\\ \scriptsize \; or \; 0.20497 \)

iv) \( \sqrt [4] {\frac {0.078 }{0.652 \; \times \; 0.841}} \)

Number		Logarithm
0.078		\(\scriptsize \bar {2}.8921 \rightarrow \)		\(\scriptsize \bar {2}.8921 \)
0.652		\( \scriptsize \bar{1}.8142\)
0.841	+	\( \scriptsize \bar{1}.9248 \)
		\(\scriptsize\bar{1}.7390 \rightarrow \)	–	\(\scriptsize \bar{1}.7390 \)
				\(\scriptsize\bar{1}.1531 \; \div \; 4\)
				\(\scriptsize (\bar{4} + 3.1531 )\; \div \; 4\)
				\(\scriptsize\bar{1}.7883 \)
				Antilog = 6.142
				Answer = \( \scriptsize 6.142 \times 10^{-1} \\ \scriptsize \: or \: 0.642 \)