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## SS2: MATHEMATICS - 1ST TERM

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Lesson 5, Topic 6
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# Calculations using Logarithms

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#### Topic Content:

• Calculations using Logarithms
• Multiplication and Division
• Power & Roots

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

1st 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 $$\scriptsize \rightarrow 2.435$$

2nd Method

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2. $$\scriptsize 912.4\: \div \: 53.55$$

Using the tables

1st 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 $$\scriptsize \rightarrow 1.704$$

2nd Method

### 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

1st 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

2nd Method

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

Solution

1st 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

2nd Method

### 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

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

Solution

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