Solve exactly: log(x+6)-log(x)=1

Answer 1

log(x+6)-log(x)

= log((x+6)/x) = 1

(x+6)/x = 10

10x = x+6

9x = 6

x =6/9 = 2/3

x=2/3

check the solution, plut 2/3 into the original equation

log(2/3 +6) + log(2/3)

= .824 + .176

= 1

when the base is not specified expressly it usully means the common log with base 10

but in more advanced mathematics, it's used as the natural log

then

ln(x+6)-ln(x)

= ln((x+6)/x)=1

x+6)/x = e

x+6 =ex

(1-e)x =-6

x =-6/(1-e)

x = exactly 6/(e-1)

x= about 3.49

check the answer

ln(3.49+6) -ln(3.49)

=2.25 -1.25

=1