IF log 2 , log (2 to the power x - 1) and log (2x+3) are in A.P. Find x

Asked by nikitaayyagari | 19th Nov, 2015, 12:58: AM

Expert Answer:

G i v e n space t h a t comma space log 2 comma space log open parentheses 2 to the power of x minus 1 close parentheses comma space log open parentheses 2 to the power of x plus 3 close parentheses space a r e space i n space A P. T h e r e f o r e comma space w e space h a v e comma l o g open parentheses 2 to the power of x minus 1 close parentheses equals fraction numerator l o g 2 plus l o g open parentheses 2 to the power of x plus 3 close parentheses over denominator 2 end fraction rightwards double arrow 2 l o g open parentheses 2 to the power of x minus 1 close parentheses equals l o g 2 plus l o g open parentheses 2 to the power of x plus 3 close parentheses rightwards double arrow l o g open parentheses 2 to the power of x minus 1 close parentheses squared equals l o g 2 plus l o g open parentheses 2 to the power of x plus 3 close parentheses space space space space space space open square brackets because a log b equals log b to the power of a close square brackets rightwards double arrow l o g open parentheses 2 to the power of x minus 1 close parentheses squared equals l o g open square brackets 2 open parentheses 2 to the power of x plus 3 close parentheses close square brackets space space space space space space space space space space space open square brackets because log a plus log b equals log a b close square brackets rightwards double arrow open parentheses 2 to the power of x minus 1 close parentheses squared equals 2 open parentheses 2 to the power of x plus 3 close parentheses space space rightwards double arrow open parentheses y minus 1 close parentheses squared equals 2 open parentheses y plus 3 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets a s s u m i n g space 2 to the power of x equals y close square brackets space rightwards double arrow y squared plus 1 minus 2 y equals 2 y plus 6 rightwards double arrow y squared plus 1 minus 2 y minus 2 y minus 6 equals 0 rightwards double arrow y squared minus 4 y minus 5 equals 0 rightwards double arrow y squared minus 5 y plus y minus 5 equals 0 rightwards double arrow y open parentheses y minus 5 close parentheses plus 1 open parentheses y minus 5 close parentheses equals 0 rightwards double arrow open parentheses y plus 1 close parentheses open parentheses y minus 5 close parentheses equals 0 rightwards double arrow y plus 1 equals 0 space o r space y minus 5 equals 0 rightwards double arrow y equals negative 1 space o r space y equals 5  N e g l e c t i n g space t h e space n e g a t i v e space a n s w e r comma space w e space h a v e comma y equals 5 rightwards double arrow 2 to the power of x equals 5 rightwards double arrow log subscript 2 2 to the power of x equals log subscript 2 5 rightwards double arrow x l o g subscript 2 2 equals l o g subscript 2 5 rightwards double arrow x equals l o g subscript 2 5  space space

Answered by Vimala Ramamurthy | 19th Nov, 2015, 09:04: AM