Algebra( solving logarithmic equations), Lecture notes of Mathematics

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Lecture 2

Logarithms

A logarithm or simply log for short is

another name for a power or index.

When the base of a logarithm is not

written it, means it is in base 10.

The logarithms of a negative number

does not exist.

Key Notes

log 2 8 = 3

1. W 𝑟𝑖𝑡𝑒 2 3 = 8 𝑖𝑛 𝑙𝑜𝑔𝑎𝑟𝑖𝑡h𝑚𝑖𝑐 𝑓𝑜𝑟𝑚.

2 .𝑊𝑟𝑖𝑡𝑒 4 2 = 16 𝑖𝑛𝑙𝑜𝑔𝑎𝑟𝑖𝑡h𝑚𝑖𝑐 𝑓𝑜𝑟𝑚.

log 4 16 = 2

log 5 1 = 0

1. W 𝑟𝑖𝑡𝑒 5 0 = 1 𝑖𝑛 𝑙𝑜𝑔𝑎𝑟𝑖𝑡h𝑚𝑖𝑐 𝑓𝑜𝑟𝑚.

Rule 1 𝐥𝐨𝐠 𝒃 𝒎 + 𝐥𝐨𝐠 𝒃 𝒏 = 𝐥𝐨𝐠 𝒃 𝒎𝒏 1_._ log 2 4 +log 2 3 =log 2 ( 4 × 3 )=log 2 12

2. log 3 5 + log 3 7 =log 2 ( 5 × 7 )=log 2 35
3. log 5 11 + log 2 9 =log 2 ( 11 × 9 )=log 2 99 𝐺𝑖𝑣𝑒𝑛𝑡h𝑎𝑡 𝑚 𝑎𝑛𝑑 𝑛 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒

Rule 2

𝒃

𝒃

𝒃 (^) (

)

1. log 10 15− log 10 5= lo g (^10) ( 15 5 ) = lo g 10 3
2. log 3 70− lo g 3 10= lo g (^3) ( 70 10 ) = lo g 3 7
3. lo g 5 12− lo g 5 3= lo g (^5) ( 12 3 ) = log 5 4

Rule 4 𝐥𝐨𝐠 𝒃 𝒎 = 𝐥𝐨𝐠 𝒂 𝒎 𝐥𝐨𝐠 𝒂 𝒃

1. log 2 8 = log 5 7 log 5 2
2. log 7 12 = log 3 12 log 3 7
3. log 3 4 = log 2 4 log 2 3

Rule 5 𝐥𝐨𝐠 𝒃 𝒃 = 𝟏

2. log 4 4 = 1
3. log 7 7 = 1
4. log 9 9 = 1

Illustrative examples

Example 1 log 3 24 + log 3 15 − log 3 10 =log 3 ( 24 × 15 ) − log 3 10 ¿ log 3 360 − log 3 10 ¿ log (^3) ( 360 10 ) ¿ log 3 36 log 3 24 + log 3 15 − log 3 10 Simplify Solution

Example 2 Expand log ( 10 𝑥 4 3 𝑦 (^2) )

log ( 10 𝑥 4 3 𝑦 (^2) ) log 10 𝑥 4 − log 3 𝑦 2 log 10 + log 𝑥 4 − (log 3 + log 𝑦 2 ) log 10 + 4 log 𝑥 − log 3 − 2 log 𝑦

Example 3 log 3 ( 9 𝑥 4 √ 𝑦^ ) =log 3 9 𝑥 4 − log 3 𝑦 1 2 ¿ log 3 9 +log 3 𝑥 4 − log 3 𝑦 1 2 ¿ log 3 3 2

• 4 log 3 𝑥 − 1 2 log 3 𝑦 ¿ 2 log 3 3 + 4 log 3 𝑥 − 1 2 log 3 𝑦 ¿ 2 + 4 log 3 𝑥 − 1 2 log 3 𝑦 Write the term log^3 ( 9 𝑥 4 √ 𝑦^ ) . 𝑺𝒐𝒍𝒖𝒕𝒊𝒐𝒏

Solution (b)

Example 4 Con’t

Trial Questions