Exercise algebra.manipulation.exponential.equation

2^x == 2^4*sqrt 2

2^x == 2^4*2^(1/2)

⇒ algebra.manipulation.exponents.equation.same-base

2^x == 2^(9/2)

x == 9/2

2.

2^(x+2) == 1/4

2^(x+2) == 1/2^2

2^(x+2) == (2^2)^(-1)

⇒ algebra.manipulation.exponents.equation.same-base

2^(x+2) == 2^(-2)

x+2 == -2

⇒ algebra.equations.coverup.onevar.plus

x == -4

3.

3^(x-1) == 81

⇒ algebra.manipulation.exponents.equation.same-base

3^(x-1) == 3^4

x-1 == 4

x == 5

4.

3^(x+5) == 243/sqrt 3

3^(x+5) == 3^5/sqrt 3

3^(x+5) == 3^5*sqrt 3^(-1)

3^(x+5) == 3^5*(3^(1/2))^(-1)

3^(x+5) == 3^5*3^(-1/2)

⇒ algebra.manipulation.exponents.equation.same-base

3^(x+5) == 3^(9/2)

x+5 == 9/2

⇒ algebra.equations.coverup.onevar.plus

x == -1/2

5.

5^(2-x) == 1/25

5^(2-x) == 1/5^2

5^(2-x) == (5^2)^(-1)

⇒ algebra.manipulation.exponents.equation.same-base

5^(2-x) == 5^(-2)

2-x == -2

x == 4

6.

3^(2*x) == 1/9

3^(2*x) == 1/3^2

3^(2*x) == (3^2)^(-1)

⇒ algebra.manipulation.exponents.equation.same-base

3^(2*x) == 3^(-2)

2*x == -2

x == -1

7.

3^(1-3*x) == 81

⇒ algebra.manipulation.exponents.equation.same-base

3^(1-3*x) == 3^4

1-3*x == 4

3*x == -3

x == -1

8.

3^(3*x-2) == 3*sqrt 3

⇒ algebra.manipulation.exponents.simple-add-exponents

3^(3*x-2) == 3*3^(1/2)

3^(3*x-2) == 3^(3/2)

⇒ algebra.manipulation.exponents.equation.same-base

3*x-2 == 3/2

3*x == 7/2

x == 7/6

9.

5*2^(x-1) == 20*sqrt 2

2^(x-1) == 4*sqrt 2

2^(x-1) == 2^2*sqrt 2

2^(x-1) == 2^2*2^(1/2)

⇒ algebra.manipulation.exponents.equation.same-base

2^(x-1) == 2^(5/2)

x-1 == 5/2

x == 7/2

10.

6*5^(2-x) == 150

5^(2-x) == 25

⇒ algebra.manipulation.exponents.equation.same-base

5^(2-x) == 5^2

2-x == 2

x == 0

11.

2*7^(4*x-1) == 98

7^(4*x-1) == 49

⇒ algebra.manipulation.exponents.equation.same-base

7^(4*x-1) == 7^2

4*x-1 == 2

4*x == 3

x == 3/4

12.

8*3^(5-2*x) == 72*sqrt 3

3^(5-2*x) == 9*sqrt 3

3^(5-2*x) == 3^2*sqrt 3

3^(5-2*x) == 3^2*3^(1/2)

⇒ algebra.manipulation.exponents.equation.same-base

3^(5-2*x) == 3^(5/2)

5-2*x == 5/2

2*x == 5/2

⇒ algebra.equations.coverup.minus-left

x == 5/4

13.

2^x-7 == 9

2^x == 16

⇒ algebra.manipulation.exponents.equation.same-base

2^x == 2^4

x == 4

14.

4^(3*x)+5 == 69

⇒ algebra.equations.coverup.plus

4^(3*x) == 64

(2^2)^(3*x) == 64

(2^2)^(3*x) == 2^6

⇒ algebra.manipulation.exponents.equation.same-base

2^(6*x) == 2^6

6*x == 6

x == 1

15.

7*3^(2*x+1) == 189

3^(2*x+1) == 27

⇒ algebra.manipulation.exponents.equation.same-base

3^(2*x+1) == 3^3

2*x+1 == 3

⇒ algebra.equations.coverup.onevar.plus

2*x == 2

x == 1

16.

5*2^(1-4*x)+11 == 51

⇒ algebra.equations.coverup.plus

5*2^(1-4*x) == 40

2^(1-4*x) == 8

⇒ algebra.manipulation.exponents.equation.same-base

2^(1-4*x) == 2^3

1-4*x == 3

4*x == -2

x == -1/2

17.

5^(x-4) == (1/5)^(2*x+1)

5^(x-4) == (5^(-1))^(2*x+1)

⇒ algebra.manipulation.exponents.equation.same-base

5^(x-4) == 5^(-2*x-1)

x-4 == -2*x-1

⇒ algebra.equations.linear.var-left, term=(-2)*x

3*x-4 == -1

3*x == 3

⇒ algebra.manipulation.exponents.equation.equals-one

x == 1

18.

7^(1-2*x) == 1

1-2*x == 0

2*x == 1

x == 1/2

19.

4^(2*x-3) == 2*sqrt 2

(2^2)^(2*x-3) == 2*sqrt 2

(2^2)^(2*x-3) == 2*2^(1/2)

⇒ algebra.manipulation.exponents.simple-add-exponents

2^(2*(2*x-3)) == 2*2^(1/2)

2^(2*(2*x-3)) == 2^(3/2)

⇒ algebra.manipulation.exponents.equation.same-base

2*(2*x-3) == 3/2

⇒ algebra.equations.linear.distr-times

4*x-6 == 3/2

4*x == 15/2

x == 15/8

20.

2*9^(1-2*x) == 6*sqrt 3

9^(1-2*x) == 3*sqrt 3

(3^2)^(1-2*x) == 3*sqrt 3

(3^2)^(1-2*x) == 3*3^(1/2)

⇒ algebra.manipulation.exponents.simple-add-exponents

3^(2*(1-2*x)) == 3*3^(1/2)

3^(2*(1-2*x)) == 3^(3/2)

⇒ algebra.manipulation.exponents.equation.same-base

2*(1-2*x) == 3/2

⇒ algebra.equations.linear.distr-times

2-4*x == 3/2

4*x == 1/2