Solve the following logarithmic equation

Question

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

Answer 1

Answer: x=7, 1

First, use the properties of logarithms to move the 2 up front. It can become an exponent:

log₃(x-4)² + log₃3=3

Use properties of logarithms again to combine the two logs, since they have the same base

When two logs of the same base are being added, you can simplify them to multiplication

log₃(x-4)²(3)=3

Another property, is that you can take the base, to the power of the number the equation is equal to, and then set that equal to the rest. So..

3³=3(x-4)² Now solve for x

27=3(x-4)(x-4) Foil the x-4's

27=3(x²-8x+16) Distribute the 3

27=3x²-24x+48 Subtract 27

0=3x²-24x+21 Factor out a 3

3(x²-8x+7)=0 Factor the inside; what two #'s multiply to get 7 and add to get -8

(x-7)(x-1)

x=7, 1

Check your work to make sure both answers satisfy the original equation.

Answer 2

2 log₃ (x - 4) + log₃ 3 = 3

log₃ (x - 4)² + log₃ 3 = 3

log₃ 3(x - 4)² = 3

3³ = 3(x - 4)²

Can you solve for x and answer?