Solve 64x 1256 Show work Solve log33x 4 log3x 1 2 Show

Solve: 64^x = 1/256 Show work. Solve: log_3(3x + 4) - log_3(x + 1) = 2 Show work for solving and checking.

Solution

5) 64^x = 1/256

==> (2⁶)^x = 1/2⁸

==> 2^6x = 2^-8          since 1/aⁿ = a^-n ; (a^m)ⁿ = a^mn

Since the bases are equal the powers can be equated

==> 6x = -8

==> x = -8/6

==> x = -4/3

Hence x = -4/3

6) log₃ (3x +4) - log₃ (x + 1) = 2

==> log₃ [(3x +4)/(x +1)] = 2           since loga - logb = log(a/b)

==> (3x +4)/(x +1) = 3²         since if log_b x =a then x = b^a

==> (3x +4)/(x +1) = 9

==> 3x +4 = 9(x +1)

   ==> 3x +4 = 9x + 9

==> 9x -3x = 4 -9

==> 6x = -5

==> x = -5/6

Hence log₃ (3x +4) - log₃ (x + 1) = 2 ==> x = -5/6

Checking log₃ (3x +4) - log₃ (x + 1) = 2

Lhs = log₃ (3x +4) - log₃ (x + 1)

==> log₃ (3(-5/6) +4) - log₃ ((-5/6) + 1)

==> log₃ (-5/2 +4) - log₃ (-5/6 + 1)

==> log₃ (3/2) - log₃ (1/6)

==> log₃ 3 - log₃ 2 - (log₃ 1 - log₃ 6)      since log (a/b) = loga - logb

==> 1 - log₃ 2 - 0 + log₃ (2*3)          since log_a a = 1 , log 1 = 0

==> 1 - log₃ 2 - 0 + log₃ 2 + log₃ 3      since log(ab) = loga + logb

==> 1 + log₃ 3

==> 1 +1

==> 2

==> Rhs

Hence log₃ (3x +4) - log₃ (x + 1) = 2