What is solution of the equation : log(4x+16) - log100 = log16 - 2*log2
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The equation to be solved is: log(4x+16) - log100 = log16 - 2log2
Use the relations: log a - log b = log(a/b), log + log b = log a*b and n*log a = log a^n
log(4x+16) - log100 = log16 - 2log2
=> log(4x+16) = log100 + log16 - log 2^2
=> log(4x+16) = log 100 + log 16 - log 4
=> log(4x+16) = log 100*16/4
=> log(4x+16) = log 400
We can equate 4x + 16 = 400
=> x + 4 = 100
=> x = 96
The required value is x = 96
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We'll apply quotient rule, both sides of the equation:
log [(4x+16)/100] = log(16/4)
Since the bases are matching, we'll apply one to one property:
[(4x+16)/100] = 16/4
[(4x+16)/100] = 4
We'll multiply by 100 both sides:
4x + 16 = 4*100
We'll divide by 4:
x + 4 = 100
We'll subtract 4:
x = 100 - 4
x = 96
The constraint of existence of logarithm is 4x+16>0
4x>-16
x>-4
The interval of admissible values for x is (-4 ; +infinite).
Since the value of x is in the rangle of admissible values, we'll accept as solution of the equation x = 96.
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