**Multiplication and Division of number greater than one using logarithm**

To multiply and divide numbers using logarithms, first express the number as logarithm and then apply the addition and subtraction laws of indices to the logarithms. Add the logarithm when multiplying and subtract when dividing.

Examples: Evaluate using logarithm.

- 4627 x 29.3
- 8198 ÷ 3.905
- 48.63 x 8.53

15.39

**Solutions**

- 4627 x 29.3

To find the Antilog of the log 5.1322 use the antilogarithm table:

Check 13 under 2 diff 2 (add the value of the difference) the number is 0.1356. To place the decimal point at the appropriate place, add one to the integer of the log i.e. 5 + 1 = 6 then shift the decimal point of the antilog figure to the right (positive) in 6 places.

- 819.8 x 3.905

** No Log**

819.8 2.9137

3.905 0.5916

antilog → 209.9 2.3221

** therefore** 819.8 ÷ 3.905 = __209.9__

- 48.63 X 8.8.53

15.39

**EVALUATION (Use the box at the bottom to post your answer for discussion and appraisal):**

- Use table to find the complete logarithm of the following:

(a) 183 (b) 89500 (c) 10.1300 (d) 7

2 Use logarithm to calculate.

3612 x 750.9

113.2 x 9.98

**Using logarithm to solve problems with powers and root**

**Simple example for powers: source here**

We seek to calculate 2345

Using rule (3), log(2345)=345∗log2

We already memorized that log2=0.30103 so this is 345∗0.30103=103.85535

Therefore using rule (5), 2345=10103.85535

We can simplify this with rule (1) to 2345=100.85535∗10103

Using the algorithm for estimating antilogarithms, we calculate that 100.85535=7.1672

Therefore our answer is 7.1672∗10103

**More difficult example for powers:**

We attempt the impossible-looking ππ. log(ππ)=π∗logπ

Using our algorithm for calculating logarithms we approximate logπ as follows:

log3.14159=log227–0.04%=log11+log2–log7–0.00432∗0.04

Note: the 0.04% is from quickly approximating 227–ππ

log3.14159=1.04139+0.30103–0.84510–0.00017=0.49715

The most difficult part in this calculation is the blind multiplication of 3.14159 and 0.49715 to 5 decimal places:

log(ππ)=3.14159∗0.49715=1.56184

Then again we use our method for antilogarithms to compute ππ=101.56184

Try 36: log36=1.55630 so ππ=36∗100.00554

0.00554/0.00432=1.28 so ππ=36+1.28%=36.46

**Example for roots:**

The same method works for any roots – except that we perform a division rather than a multiplication. As an example we will calculate 902.54−−−−−√7

Using the method for calculating logarithms, log902.54=2.95547

log902.54−−−−−√7=log902.547=2.955477=0.42221 902.54−−−−−√7=100.42221

Using the method for calculating antilogarithms, 100.42221=2.643 and this is our answer.

**EVALUATION (Use the box at the bottom to post your answer for discussion and appraisal)**

- Find the logarithm of the following:

(a) 0.064 (b) 0.002 (c) 0.802

- Evaluate using logarithm.

95.3 x √ 318.4

1.29^{5} x 2.03

**Using logarithm to evaluate problems of Multiplication, Division, Powers and roots with numbers less than one.**

Examples:

- 0.6735 x 0.928
- 0.005692 ¸ 0.0943
- 0.6104
^{3} - 4 √ 0.000

- 3 √ 0.06642

Solution

- 0.6735 x 0.928

__No. Log._____

0.6735 1.8283

0.928 1.9675

**0.6248 1.7958**

**Therefore 0.6735 x 0.928 = 0.6248**

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