Calculations using Logarithms

The basic principles of calculation using logarithms depend on the laws of indices.

Example 5.6.1 – Multiplication and Division:

Use tables to work out the following:

1. 26.52 × 9.184
2. 912.4  ÷ 94.35

Solution

1. 26.52 × 9.184

1^st Method

Using the tables

i.e \( \scriptsize 10^{1.4235} \: \times \: 10^{0.9630} \)

= \( \scriptsize 10^{1.4235 \:+ \:0.9630} \rightarrow \left(x^a \: \times \: x^b = x^{a \: + \:b} \right) \)

= \( \scriptsize 10^{2.3865}\)

Using the anti-log \( \scriptsize \rightarrow 2.435 \)

Answer = 243.5

2nd Method

No	Log
26.52	1.4235
9.184	+ 0.9630
	2.3865
	Antliog of 0.3865 = 2.435
	102 × 2.435
	Ans = 243.5

2. \( \scriptsize 912.4\: \div \: 53.55 \)

Using the tables

1^st Method

i.e \( \scriptsize 10^{2.9602} \: \div \: 10^{1.7288} \)

= \( \scriptsize 10^{2.9602 \:- \: 1.7288} \rightarrow \left(x^a \: \div \: x^b = x^{a \: – \: b} \right) \)

= \( \scriptsize 10^{1.2314} \) Using the anti-log \( \scriptsize \rightarrow 1.704\)

Answer = 17.04

2nd Method

No	Log
912.4	2.9602
53.55	–1.7288
	1.2314
	Antliog of 0.2314 = 1.704
	101 x 1.704
	Ans = 17.04

Power & Roots:

Example 5.6.2:

Use logarithms to evaluate the following:

(i) \( \scriptsize \left (3.9562 \right)^3 \)
(ii) \( \scriptsize \sqrt [6] { 68.15} \)

(i) \( \scriptsize \left (3.9562 \right)^3 \)

Solution

1^st Method

= \( \scriptsize \left (10^{0.5973} \right)^3 \rightarrow \left (a^x \right)^y = a^{xy} \)

= \( \scriptsize 10^{1.7919} \rightarrow \; using \; antilog \; \)

= 6.193

Answer = 61.93

2^nd Method

No	Log
\( \scriptsize \left (3.9562 \right)^3 \)	0.5973
	0.5793 × 3
	= 1.7919
	Antilog of 0.7919 = 6.193
	101 × 6.193
	Ans = 61.93

(ii) \( \scriptsize \sqrt [6] { 68.15} \)

Solution

1^st Method

\( \scriptsize \sqrt [6] { 68.15} = \left (68.15 \right)^{\frac{1}{6}} \)

= \( \scriptsize 10^{1.8334 \: \div \: 6} \rightarrow \left ( \sqrt [x] {a} = a ^{ \frac{1}{x}} \right) \)

= \( \scriptsize \left (10^{0.3056} \right) \) Using antilog

= 2.021

  Answer =  2.021

2^nd Method

No	Log
\( \scriptsize \sqrt [6] { 68.15} \)
\( \scriptsize \left( 68.15 \right)^{\frac{1}{6}} \)	1.8335 ÷ 6
	= 0.3056
	Antilog of 0.3056 = 2.021
	100 × 2.021
	Ans = 2.021

Example 5.6.3:

Use the log tables to evaluate the following:

(i) \( \frac{(18.6)^2 \: \times \: 9.76}{\sqrt[4]{8500}} \)

(ii) \( \sqrt[4]{\left[ \frac{43.12 \: \times \: 4.08}{3.401 \: \times \: 2.184}\right]^3} \)

(i) \( \frac{(18.6)^2 \: \times \: 9.76}{\sqrt[4]{8500}} \)

Solution

No	Log
\( \scriptsize(18.6)^2 \)	1.2695 × 2 =		2.5390
9.76	0.9894	+	0.9894
			3.5284
\(\scriptsize \sqrt[4]{(8500)} \)
\( \scriptsize \left(8500 \right)^{\frac{1}{4}}\)	3.9294 ÷ 4 =	–	0.9824
			2.5460
			Antilog of 0.5460 = 3.516
			102 × 3.516
			Ans = 351.6

(ii) \( \sqrt[4]{\left[ \frac{43.12 \: \times \: 4.08}{3.401 \: \times \: 2.184}\right]^3} \)

Solution

No		Log
43.12		1.6347
4.08	+	0.6107
		2.2454	→		2.2454
3.401	+	0.5316
2.184		0.3393
		0.8709	→	–	0.8709
				\(\scriptsize \left[ 1.3745 \right]^3\) =	1.3745 × 3
				\( \scriptsize\sqrt[4]{4.1235}\) =	4.1235 ÷ 4
				=	1.0309
					Antilog of .0309 = 1.0737
					101 × 1.0737
					Ans = 10.737
					= 10.74 (2.dp)