20 solved step by step problems :Logarithm and exponentials.

LOGARITHM AND EXPONENTIALS. HOW TO WRITE AN EXPRESSION AS A LOGARITHM log􀯔 􀝊 􀵌 􀝔 means that 􀜽􀯫 􀵌 􀝊, where a is called the base of the logarithm EXAMPLES Here a = 2, 􀝔 􀵌 5 􀜽􀝊􀝀 􀝊 􀵌 32 2􀬹=32 So, log􀬶 32 􀵌 5 ( Match the position of respective colours) logarithm base In other words you would say ‘ 2 to the power 5 equals 32’ Rewrite as logarithm Example 1 10􀬷 􀵌 1000 log􀬵􀬴 1000 􀵌 3 Example 2 5􀬸 􀵌 625 log􀬹 625 􀵌 4 Example 3 2􀬵􀬴 􀵌 1024 log􀬶 1024 􀵌 10 Example 4 log􀯔 1 􀵌 0 (because) 􀜽􀬴 􀵌 1 Example 5 log􀯔 􀜽 􀵌 1(because) 􀜽􀬵 􀵌 􀜽 Find the value of Example 6: log􀬷 81 􀵌? Let log􀬷 81 􀵌 􀝔 Therefore 3􀯫= 81 3􀯫 􀵌 3􀬸 ( because 3􀬸 􀵌 3 􀵈 3 􀵈 3 􀵈 3 􀵌 81􁈻 􀝔 􀵌 4 Therefore, log􀬷 81 􀵌 4 􀜣􀝊􀝏. (when bases are same powers would be equal, if powers are equal bases would be same) Example 7: log􀬸 0.25 􀵌? Let log􀬸 0.25 􀵌 􀝔 4􀯫 􀵌 0.25 4􀯫 􀵌 􀬵􀬸 ( because 0.25=􀬵􀬸) 4􀯫 􀵌 4􀬿􀬵 􀝔 􀵌 􀵆1 Therefore log􀬷 81 􀵌 􀵆1 Ans. Example 8: log􀯔 􀜽􀬹 􀵌? Let log􀯔 􀜽􀬹 =􀝔 􀜽􀯫 􀵌 􀜽􀬹 􀝔 􀵌 5 Therefore, log􀯔 􀜽􀬹 􀵌 5 Ans. FIND THE VALUE OF X Example 9: 10􀯫 􀵌 500 Applying log on both side, we get log􀬵􀬴 10􀯫 􀵌 log􀬵􀬴 500 􀝔 log􀬵􀬴 10 􀵌 log􀬵􀬴 500 (because log􀯔 􀜿􀯗 􀵌 dlog􀯔 􀜿) 􀝔􁈺1􁈻 􀵌 log􀬵􀬴 500 (because log􀬵􀬴 10 􀵌 1􁈻 􀝔 􀵌 log􀬵􀬴 500 􀵌 2.70 (using calculator or log tables) LAWS OF LOGARITHMS MULTIPLICATION LAW log􀯔 􀝔􀝕 􀵌 log􀯔 􀝔 􀵅 log􀯔 􀝕 You can also say that when log applied on multiplication or product it would convert it, into sum of individual logs. Suppose that log􀯔 􀝔 􀵌 􀜾 􀜽􀝊􀝀 log􀯔 􀝕 􀵌 􀜿 Rewriting with powers 􀜽􀯕 􀵌 􀝔 􀜽􀝊􀝀 􀜽􀯖 􀵌 􀝕 Multiplying the two equations : 􀝔􀝕 􀵌 􀜽􀯕􀜽􀯖 􀝔􀝕 􀵌 􀜽􀯕􀬾􀯖 Rewriting as logarithm log􀯔 􀝔􀝕 􀵌 􀜾 􀵅 􀜿 log􀯔 􀝔􀝕 􀵌 log􀯔 􀝔 􀵅 log􀯔 􀝕 DIVISION LAW log􀯔 􁉀􀯫􀯬􁉁 􀵌 log􀯔 􀝔 􀵆 log􀯔 􀝕 POWER LAW log􀯔􁈺􀝔􁈻􀯞 􀵌 􀝇 log􀯔 􀝔 WRITE AS A SINGLE LOGARITHM EXAMPLE 10 log􀬷 6 􀵅 log􀬷 7=? SOLUTION log􀬷 6 􀵅 log􀬷 7 􀵌 log􀬷􁈺6 􀵈 7􁈻 ( using multiplication law) 􀵌 log􀬷 42 EXAMPLE 11 log􀬶 15 􀵆 log􀬶 3 SOLUTION log􀬶 15 􀵆 log􀬶 3 􀵌 log􀬶 􀬵􀬹 􀬷 ( using division law) 􀵌 log􀬶 􀬵􀬹 􀬷 􀵌 log􀬶 5 EXAMPLE 12 2 log􀬹 3 􀵅 3 log􀬹 2 SOLUTION First apply the power law to both the parts and then use the multiplication law 2 log􀬹 3 􀵌 log􀬹􁈺3􁈻􀬶 􀵌 log􀬹 9 (power law) 3 log􀬹 2 􀵌 log􀬹􁈺2􁈻􀬷 􀵌 log􀬹 8 Therefore, 2 log􀬹 3 􀵅 3 log􀬹 2 􀵌 log􀬹 9 􀵅 log􀬹 8 􀵌 log􀬹􁈺9 􀵈 8􁈻 (multiplication law) 􀵌 log􀬹 72 EXAMPLE 13 log􀬵􀬴 3 􀵆 4 log􀬵􀬴􁈺12􁈻 SOLUTION Use first the power law and then division law 4 log􀬵􀬴 12 􀵌 log􀬵􀬴􁈺12􁈻􀬸 􀵌 log􀬵􀬴 􀬵 􀬵􀬺 Therefore, log􀬵􀬴 3 􀵆 4 log􀬵􀬴􁈺12􁈻 􀵌 log􀬵􀬴 3 􀵆 log􀬵􀬴 1 16 􀵌 log􀬵􀬴 􀬷􀰭 􀰭􀰲 ( Division Law) 􀵌 log􀬵􀬴 48 Ans. ( 􀬷􀰭 􀰭􀰲 􀵌 3 􀵊 􀬵 􀬵􀬺 􀵌 3 􀵈 􀬵􀬺 􀬵 􀵌 48) WRITE IN TERMS OF 􀜔􀜗􀜏􀢇 􀢞 , 􀜔􀜗􀜏􀢇 􀢟 , 􀜔􀜗􀜏􀢇 􀢠 EXAMPLE 14 log􀯔 􀝔􀬶􀝕􀝖􀬷 SOLUTION log􀯔 􀝔􀬶􀝕􀝖􀬷 􀵌 log􀯔 􀝔􀬶 􀵅 log􀯔 􀝕+log􀯔 􀝖􀬷 􀵌 2 log􀯔 􀝔 􀵅 log􀯔 􀝕 􀵅 3log􀯔 􀝖 Ans. EXAMPLE 15 log􀭟􁈺 􀝔 y􀬷􁈻 SOLUTION log􀭟􁈺 􀯫 􀭷􀰯􁈻 = log􀯔 􀝔 􀵆 log􀯔 􀝕􀬷 = log􀯔 􀝔 􀵆 3 log􀯔 􀝕 EXAMPLE 16 log􀭟x􀶥y z SOLUTION log􀭟􀭶􀶥􀭷 􀭸 􀵌 log􀯔 􀝔􀶥􀝕 􀵆 log􀯔 􀝖 􀵌 log􀯔 􀝔 􀵅 log􀯔 􀶥􀝕 􀵆 log􀯔 􀝖 = log􀯔 􀝔 􀵅 􀬵􀬶 log􀯔 􀝕 􀵆 log􀯔 􀝖 (b/c √ means 􀬵􀬶 and using power law) EXAMPLE 17 log􀭟􀝔 􀜽􀬸 SOLUTION log􀭟􀝔 􀜽􀬸 􀵌 log􀯔 􀝔 􀵆 log􀯔 􀜽􀬸 􀵌 log􀯔 􀝔 􀵆 4log􀯔 􀜽 􀵌 log􀯔 􀝔 􀵆 4􁈺1􁈻 􀵌 log􀯔 􀝔 􀵆 4 Ans. SOLVING EQUATIONS OF THE FORM 􀢇􀢞 􀵌 􀢈 EXAMPLE 18 Solve 3􀯫 􀵌 20 SOLUTION To solve an equation means to find the value of unknown variable , therefore here we have to find the value of 􀝔. Since your calculator has only base 10 logarithm and natural log (ln). So any working must be done in these two bases only. 3􀯫 􀵌 20 log􀬵􀬴 3􀯫 􀵌 log􀬵􀬴 20 ( Applying log on both sides) 􀝔 log􀬵􀬴 3 􀵌 log􀬵􀬴 20 ( Applying power law) 􀝔 􀵌 􀭪􀭭􀭥􀰭􀰬 􀬶􀬴 􀭪􀭭􀭥􀰭􀰬 􀬷 (dividing by log􀬵􀬴 3, to find 􀝔􁈻 􀝔 􀵌 􀬵.􀬷􀬴􀬵􀬴 􀬴.􀬸􀬻􀬻􀬵 􀝔 􀵌 2.73 Ans. EXAMPLE 19 Solve 7􀯫􀬾􀬵 􀵌 3􀯫􀬾􀬶 SOLUTION 7􀯫􀬾􀬵 􀵌 3􀯫􀬾􀬶 Applying log on b.s 􀝈􀝋􀝃7􀯫􀬾􀬵 􀵌 􀝈􀝋􀝃3􀯫􀬾􀬶 􁈺􀝔 􀵅 1􁈻􀝈􀝋􀝃7 􀵌 􁈺􀝔 􀵅 2􁈻􀝈􀝋􀝃3 􀝔􀝈􀝋􀝃7 􀵅 􀝈􀝋􀝃7 􀵌 􀝔􀝈􀝋􀝃3 􀵅 2􀝈􀝋􀝃3 􀝔􀝈􀝋􀝃7 􀵆 􀝔􀝈􀝋􀝃3 􀵌 2􀝈􀝋􀝃3 􀵆 􀝈􀝋􀝃7 􀝔􁈺􀝈􀝋􀝃7 􀵆 􀝈􀝋􀝃3􁈻 􀵌 2􀝈􀝋􀝃3 􀵆 􀝈􀝋􀝃7 􀝔 􀵌 2􀝈􀝋􀝃3 􀵆 􀝈􀝋􀝃7 􀝈􀝋􀝃7 􀵆 􀝈􀝋􀝃3 􀝔 􀵌 0.2966 Ans. Example 20: Solve 5􀬶􀯫 􀵅 7􁈺5􁈻􀯫 􀵆 30 􀵌 0 SOLUTION 􀜮􀝁􀝐 y 􀵌 5􀭶 􀜶􀝄􀝁􀝎􀝁􀝂􀝋􀝎􀝁 􀜽􀜾􀝋􀝒􀝁 􀝁􀝍 􀜿􀜽􀝊 􀜾􀝁 􀝓􀝎􀝅􀝐􀝐􀝁􀝊 􀜽􀝏 􁈺5􀯫􁈻􀬶 􀵅 7􁈺5􁈻􀯫 􀵆 30 􀵌 0 􀝕􀬶 􀵅 7􀝕 􀵆 30 􀵌 0 􀝕􀬶 􀵅 10􀝕 􀵆 3􀝕 􀵆 30 􀵌 0 􀝕􁈺􀝕 􀵅 10􁈻 􀵆 3􁈺􀝕 􀵅 10􁈻 􀵌 0 􁈺􀝕 􀵆 3􁈻􁈺􀝕 􀵅 10􁈻 􀵌 0 􀝕 􀵆 3 􀵌 0 􀝋􀝎 􀝕 􀵅 10 􀵌 0 􀝕 􀵌 3 􀝋􀝎 􀝕 􀵌 􀵆10 􁈺􀝕 􀵌 􀵆10 􀝉􀝁􀜽􀝊􀝏 5􀯫 􀵌 􀵆10, 􀝓􀝄􀝅􀜿􀝄 􀝄􀜽􀝏 􀝊􀝋 􀝏􀝋􀝈􀝑􀝐􀝅􀝋􀝊􁈻 􀜫􀝂 􀝕 􀵌 3 , 5􀯫 􀵌 3 log􀬵􀬴 5􀯫 􀵌 log􀬵􀬴 3 􀝔 log􀬵􀬴 5 􀵌 log􀬵􀬴 3 􀝔 􀵌 log􀬵􀬴 3 log􀬵􀬴 5 􀝔 􀵌 0.68 Ans