How do you solve #\log _ { 5} ( x + 52) + \log _ { 5} x = \log _ { 5} 53#?
1 Answer
Explanation:
Given:
#log_5(x+52) + log_5(x) = log_5(53)#
Method 1
Note that for any valid base
#log_b(1) = 0#
So:
#log_5(color(blue)(1)+52) + log_5(color(blue)(1)) = log_5(53) + 0 = log_5(53)#
So
Method 2
If
#log_b c + log_b d = log_b cd#
So, provided
#log_5(53) = log_5(x+52) + log_5(x)#
#color(white)(log_5(53)) = log_5((x+52)x)#
#color(white)(log_5(53)) = log_5(x^2+52x)#
Note that
#53 = x^2+52x#
Subtract
#0 = x^2+52x-53 = (x-1)(x+53)#
So
But considered as a real valued function, there is no value of
So the only valid real solution is
Complex footnote
It is possible to define
The principal value is
Note that with this definition,
#log_5(color(blue)(-53)+52) + log_5(color(blue)(-53))#
#= log_5(-1)+log_5(-53)#
#= (log_5(1)+pi/ln 5i)+(log_5(53)+pi/ln5i)#
#= log_5(53)+(2pi)/ln5i#
#!= log_5(53)#