Question
Solve the logarithmic equation. Express your solutions in exact form only. Please show all of your work.
log base 6 (64x^3+1)-log base 6 (4x+1)=1
log base 6 (64x^3+1)-log base 6 (4x+1)=1
Answers
log <sub>6</sub>[ (64x^3+1)/(4x+1) ] = 1
(64x^3+1)/(4x+1) = 6^1
(4x+1)(16x^2 - 4x + 1)/(4x+1) = 6
16x^2 - 4x - 5 = 0
x= (4 ± √336)/32
= (4 ± 4√21)/32
= (1 ± 21)/8
We have to reject the negative answer or else the log would be undefined
so
x = (1 + √21)/8
(64x^3+1)/(4x+1) = 6^1
(4x+1)(16x^2 - 4x + 1)/(4x+1) = 6
16x^2 - 4x - 5 = 0
x= (4 ± √336)/32
= (4 ± 4√21)/32
= (1 ± 21)/8
We have to reject the negative answer or else the log would be undefined
so
x = (1 + √21)/8