Solving Exponential Equations

3(2x-4)=12+2x

distribute: 6x-12=12+2x

Combine x's: 6x-2x-12=12+2x-2x

Simplify: 4x-12=12

Add 12 to both sides: 4x-12+12=12+12

Simplify: 4x=24

Divide by 4: x=6

Is there a like base?

Yes, then set the exponents = to each other

2x+10 = x-14

Subtract x: 2x-x+10= x-x-14

Simplify: x+10= -14

Subtract 10: x+10-10= -14-10

Simplify: x= -24

Are the bases the same?

Yes, then set them equal.

3x-4 = x+6

Subtract x: 3x-x-4= x-x+6

Simplify: 2x-4=6

Add 4: 2x-4+4= 6+4

Simplify: 2x=10

Divide by 2: x=5

Are the bases the same?

No, but 27 can be changed to be 3³

Thus: 3^n-8= 3³⁽ⁿ⁺⁴⁾

Set them equal and distribute: n-8= 3n+12

Subtract n: n-n-8= 3n-n+12

Simplify: -8= 2n+12

Subtract 12: -8-12= 2n+12-12

Simplify: -20= 2n

Divide by 2: -10= n

n= -10

Are the bases the same?

No, then find a common base

Note 4²= 16 and 4³=64

Thus; 4^2(x-5)= 4^3(5x-12)

Now set the exponents equal and distribute:

2x-10=15x-36

Subtract 2x: 2x-2x-10=15x-2x-36

Simplify: -10=13x- 36

Add 36: -10+36= 13x-36+36

Simplify 26=13x

Divide by 13: 2=x

x=2