How to solve for x for the respectable questions regarding logarithms?

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1 Answer
Mar 3, 2018

#color(blue)((b))x=1-ln16 orln(e/16)#
#color(blue)((c))x=log_(14)(7/e)#

Explanation:

#color(red)((1)log_(a)M=nhArrM=a^n#
#color(red)((2)log_(a)M-log_(a)N=log_(a)(M/N))#
#color(red)((3)nlog_(e)M=log_(e)M^n)#
#color(blue)((b)If,log_(2)7-xlog_(2)7=40#
#rArrlog_(2)7-log_(2)7^x=4#
Using (2),
#log_(2)(7/7^x)=4#
Using(1)
#7/7^x=2^4rArr7^(1-x)=16#
Again using(1)
#1-x=ln16rArrx=1-ln16=lne-ln16=x=ln(e/16)#
#color(blue)((c)log_(e)7-xlog_(e)14=1)#
Using (3),
#log_e7-color(red)(log_(e)14^x)=1#
Using(2),
#log_(e)(7/14^x)=1#
Using(1)#7/14^x=e^1rArr14^x=7/e#
Again using(1)
#x=log_(14)(7/e)#