Exponential+and+Logarithmic+Equations

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 * __Exponential and Logarithmic Equations__**


 * 1) a)** 4^(x+3) - 4^x = 63
 * b)** 3**log2(**x) - **log2**(x)=8
 * 2) a)** 4^(2x) = 5^(2x-1)
 * b)** **loga**(x+2) + **loga**(x-1) = **loga** (8-2x)
 * 3) a)** 3^(2x) - 5(3^x) = -6
 * b)** **log**(35-x^3) / **log**(5-x) = 3
 * 4) a)** 3^ (x^2+20) = (1/27)^(3x)
 * b)** **log5**(x-1) + **log5**(x-2) - **log5**(x+6) = 0

Note: Exponential and Logarithmic rules are useful to solve the equations above!

Sample Solution: 1) a) 4^x +4^3 - 4^x = 63 4^x (64 - 1) = 63 4^x (63) = 63 4^x = 1 4^x = 4^0
 * 4^(x+3) - 4^x = 63**
 * x=0**

b) 3**log2(**x) - **log2**(x)=8 x^2 = 2^8 x^2 = 256 x = +/- 16
 * log2** ( x^3 / x ) = 8
 * log2** (x^2) =8
 * final answer: x=16**
 * because negative log is impossible**

2) a) 1st method

4^(2x) = 5^(2x-1) 16^x = 5^2x / 5 16^x = 25^x/ 5 5= (25/16)^x 5= 1.5625^x log5/log1.5625 = x
 * log x= 3.61**

2nd method 4^(2x) = 5^(2x-1) log (4^2x) = log (5^ (2x-1) ) (2x)log (4) = (2x-1) log(5) (2x)(0.60) = (2x-1) (0.699) 2x = 2.32x - 1.16
 * log x= 3.61

b) loga**(x+2) + **loga**(x-1) = **loga** (8-2x) (x+2)(x-1)= (8-2x) x^2 + 3x -10 = 0 0 = (x+5) (x-2)
 * loga**(x+2) (x-1)= **loga**(8-2x)
 * Final answer: x=2**