Logarithms

Logarithms seem very weird and difficult. But if we study it with a little concentration and tricks, they are very easy too.   Let's do this, 1st of all let's divide the log into types:   1. Basic Logs:   Example 1: Log3 9 <=> Log3 9 = x <=> 3 raised to power x = 9 <=> 3*3*3 = 9 <=> x = 3 Example 2: Log 10000 <=> Log10 10000 <=> Log 10 10000 = x <=> 10 raised to power x = 10000 <=> 10*10*10*10 = 10000 <=> x = 4   2. Weird Logs:   Example 1: Log2 (1/8) <=> Log2 (1/8) = x <=> 2 raised to power x = 1/8 <=> 2 raised to power x = 1/ 2 raised to power 3 <=> 2 raised to power x = 2 raised to power -3 <=> x = -3 Example 2: Log 1 <=> Log10 1 = x <=> 10 raised to power x = 1 <=> 10 raised to power 0 = 1 <=> x = 0 Example 3: Log 0 <=> Log10 0 = x <=> 10 raised to power x = 0 <=> which is practically not possible <=> undefined Example 4: Log (-1) <=> Log10 (-1) = x <=> 10 raised to power x = -1 <=> which is practically not possible <=> undefined   3. Natural Logs:   Example 1: Ln1 <=> Loge 1 <=> Loge 1 = x <=> e raised to power x = 1 <=> e raised to power 0 = 1 <=> x = 0 Example 2: Ln (e)raised to power 3 <=> Loge (e) raised to power 3 = x <=> e raised to power x = e raised to power 3 <=> x = 3   4. Weirder Logs:   Example 1:  Logx 64 = 5 <=> x raised to power 5 = 64 <=> 2*2*2*2*2 = 64 <=> x = 5 Example 2: Log5 x = 3 <=> 5 raised to power 3 = x <=> 125 = x <=> x = 125 Example 3: Log2 7 <=> Log2 7 = x <=> 2 raised to power x = 7 <=> which we can write as Log 7/Log 2= 7     Things to remember while solving logs :  1. If there is no base of Log, then by default we take it as 10 always 2. Ln is always equaled to Log base e i.e Loge 3. If equation is 3Log2 8, it means whatever the result of log is, it will get multiplied to 3