Solve the equation

Question

log x + log(x + 30) = 3

Answer 1

logx+log(x+30)=3 (using base 10)

log(x*[x+30])=3

log(x²+30x)=3

again using base 10

10³=x²+30x

x²+30x-1000=0

factor

1000=2*2*2*5*5*5=(2*2*5)*(2*5*5)=20*50 and 50-20=30

(x+50)(x-20)=0

ignore x+50

x-20=0

x=20 (answer)

check:

log20+log50=3

log(20*50)=3

log1000=3

log(10)10³=3

3log(10)(10)=3

3*1=3

3=3

 

 

Answer 2

First combine the two log functions using sum of log

 

log(x) + log(x+30) = log(x*(x+30))

                            = log (x²+30x)

 

Which leaves you with the equation

 

log(x²+30x) = 3

 

Next raise both sides of the equation to the power of 10 and solve

 

10^log(x²+30x) = 10^3

 

since the power and the log functions cancel you are left with

 

x² + 30x = 10³ = 1000

 

x² + 30x - 1000 = 0

 

(x - 20)(x + 50) = 0

 

x = 20 or - 50

 

but since you cannot have a negative number under a log, only chose the positive answer:

 

x = 20 //