How to solve this completely?

Question

log₂x-log₂(3x+5)=4

Answer 1

We have a difference in logs.  We can rewrite this as a quotient.

 

log₂(x / (3x + 5)) = 4

 

 

The solution to a logarithm is the exponent of the log's base.

 

2⁴ = x / (3x + 5)

 

16 = x / (3x + 5)

 

 

Multiply both sides of equation by (3x + 5).

 

16(3x + 5) = x

 

48x + 80 = x

 

 

Subtract 48x on both sides of equation.

 

80 = -47x

 

 

Divide both sides of equation by -47.

 

-80/47 = x

Answer 2

log₂(x) - log₂(3x+5) = 4

 

Note that log_a(m/n) = log_am - log_a(n) so...

 

log₂ (x/(3x+5)) = 4

 

Since logarithms are the opposite of exponents defined by if aⁿ = x then log_ax=n then

 

2⁴ = x/(3x+5)

16 = x/(3x+5)

16(3x+5) = x

48x + 80 = x

47x = -80

x = -80/47