Logarithm equations

Question

Solve log base 3(2x +5)-log base 4 x =3

Answer 1

Rewrite the equation as log₃(2x + 5) − log₄x − 3 = 0.

Through logarithmic identities & Calculus, one can transform log₃(2x + 5) − log₄x − 3 = 0

to ln (2x + 5)/ln 3 − ln x/ln 4 − 3 = 0 and take the first derivative of ln (2x + 5)/ln 3 − ln x/ln 4 − 3

as [2 ÷ (2x + 5) ÷ ln 3] − [1 ÷ x ÷ ln 4].

By Newton's Method Of Root Approximation, build the Formula

x − {ln (2x + 5)/ln 3 − ln x/ln 4 − 3} ÷ [2 ÷ (2x + 5) ÷ ln 3] − [1 ÷ x ÷ ln 4].

A graphing calculator shows y = ln (2x + 5)/ln 3 − ln x/ln 4 − 3 going to 0

around x = 0.13. The table below shows evaluation of the Formula with

x initially at 0.13. Each evaluation of the formula is given back to the

Formula as the new input value of x.

x--------------------------------------------x − {ln (2x + 5)/ln 3 − ln x/ln 4 − 3} ÷ [2 ÷ (2x + 5) ÷ ln 3] − [1 ÷ x ÷ ln 4]

0.13---------------------------------------------------0.1266987741

0.1266987741--------------------------------------0.1267428895

0.1267428895--------------------------------------0.1267428977

0.1267428977--------------------------------------0.1267428977

The last line of the table gives the last evaluation of the Formula as identical to the last x input;

x = 0.1267428977 is then the value of x sought.

This value of x = 0.1267428977 gives 2x + 5 as 5.253485795.

Then equate 3^G to 5.253485795 which leads to Gln 3 = ln 5.253485795 or G = 1.509988405.

Also equate 4^P to 0.1267428977 which leads to Pln 4 = ln 0.1267428977 or P = -1.490011595.

G − P or 1.509988405 − -1.490011595 does show log₃(2x + 5) − log₄x as equal to 3.