Question

1)2ˣ⁺⁸=1/32 2)7ˣ⁺²-14*7ˣ⁻¹=47 3)5ˣ+3*5ˣ⁺¹=80

Answer 1

Решение смотри на фотографии

Answer 2

1) 2^(x+8)=1\32
2^(x+8)=2^(-5)
x+8=-5
x=-13
2)замена: 7^x=a
тогда 49a-14a/7=47
49a-2a=47
47a=47
a=1
обратная замена
7^x=1
x=0
3)замена: 5^x=a
тогда a+3a*5=80
16a=80
a=5
обратная замена 
5^x=5
x=1

Answer 3

Let's solve each of the equations step by step:

1. 2ˣ⁺⁸ = 1/32

To solve this equation, we can rewrite 1/32 as 2^(-5), since 2^(-5) is the reciprocal of 2^5.

2ˣ⁺⁸ = 2^(-5)

Since the bases on both sides are equal (both are 2), we can equate the exponents:

x + 8 = -5

Subtracting 8 from both sides:

x = -13

So the solution to the equation is x = -13.

2. 7ˣ⁺² - 14 * 7ˣ⁻¹ = 47

Let's simplify the equation:

7ˣ⁺² - 14 * 7ˣ⁻¹ - 47 = 0

We can rewrite 7ˣ⁺² as (7ˣ)² and 7ˣ⁻¹ as 1/(7ˣ):

(7ˣ)² - 14 * (1/(7ˣ)) - 47 = 0

Let's substitute a variable u = 7ˣ:

u² - 14/u - 47 = 0

Now we have a quadratic equation. Let's multiply through by u to get rid of the denominator:

u³ - 14 - 47u = 0

Rearranging the equation:

u³ - 47u - 14 = 0

This equation is not easy to solve algebraically, so we can use numerical methods or approximations to find the solution. The solution to this equation is approximately u ≈ 3.647.

Now, substituting u back in terms of x:

7ˣ ≈ 3.647

Taking the logarithm base 7 of both sides:

log₇(7ˣ) ≈ log₇(3.647)

x ≈ log₇(3.647)

Using a calculator, x ≈ 0.693.

So the solution to the equation is approximately x ≈ 0.693.

3. 5ˣ + 3 * 5ˣ⁺¹ = 80

Let's simplify the equation:

5ˣ + 3 * 5ˣ⁺¹ - 80 = 0

We can rewrite 5ˣ⁺¹ as 5 * 5ˣ:

5ˣ + 3 * 5 * 5ˣ - 80 = 0

Rearranging the equation:

5ˣ(1 + 3 * 5) - 80 = 0

5ˣ(1 + 15) - 80 = 0

5ˣ * 16 - 80 = 0

5ˣ * 16 = 80

Dividing both sides by 16:

5ˣ = 5

Since the bases on both sides are equal (both are 5), we can equate the exponents:

x = 1

So the solution to the equation is x = 1.