Log. Equations?

1)
(log(x+1))² - log(x+1)² = 8
(log(x+1))² - 2log(x+1) - 8 = 0
(log(x+1) + 2) (log(x+1) - 4) = 0
log(x+1) = - 2 或 log(x+1) = 4
x+1 = 10⁻² 或 x+1 = 10⁴
x = 1/100 - 1 = - 0.99 或 x = 10000 - 1 = 9999。

2)
5²ˣ⁺¹ = 4ˣ⁺² 3²ˣ⁻¹
5²ˣ⁺¹ = 2²ˣ⁺⁴3²ˣ⁻¹
5² (5²ˣ⁻¹) = 2⁵(2²ˣ⁻¹) 3²ˣ⁻¹
5² / 2⁵ = 6²ˣ⁻¹ / 5²ˣ⁻¹
25/32 = (6/5)²ˣ⁻¹
log(25/32) = (2x - 1)log(6/5)
2x - 1 = log(25/32) / log(6/5)
2x = (log(6/5) + log(25/32)) / log(6/5)
2x = log(6/5 * 25/32) / log(6/5)
2x = log(15/16) / log(6/5)
x = log(15/16) / (2log(6/5))
x = log(15/16) / log(36/25) = - 0.177 (準確至3位有效數字)

3)
9ˣ + 2(3ˣ⁺¹) - 16 = 0
3²ˣ + 2*3(3ˣ) - 16 = 0
(3ˣ)² + 6(3ˣ) - 16 = 0
(3ˣ - 2) (3ˣ + 8) = 0
3ˣ = 2 或 3ˣ = - 8(捨去)
xlog3 = log2
x = log2 / log3 = log₃2 = 0.631 (準確至3位有效數字)

1)
[log(x+1)]^2 -log(x+1)^2=8
[log(x+1)]^2 -2log(x+1)-8=0
log(x+1)=4 or log(x+1)=-2
x=9999 or x=-0.99

2)
5^(2X+1) =4^(X+2) * 3^(2X-1)
5[5^(2x)]=(4^2)(4^x)[3^(2x)]/3
(15/16)[5^(2x)]=(4^x)[3^(2x)]
15/16=[2^(2x)][3^(2x)]/[5^(2x)]
15/16=(6/5)^(2x)
15/16=(36/25)^x
log(15/16)=x log(36/25)
x=log(15/16)/log(36/25)
x=-1.76990922

3)
9^x + 2[3^(x+1) ]-16 =0
3^(2x)+6(3^x)-16=0
3^x=2 or 3^x=-8(捨去)
x=0.630929753