Question

(1/2)^x+1+(1/2)^x-2+(1/2)^x+3=37 Решите уравнение

Answer 1

Решение на фотографии

Answer 2

To solve the equation (1/2)^x+1 + (1/2)^x-2 + (1/2)^x+3 = 37, we can simplify the terms first.

Notice that each term on the left side of the equation contains (1/2)^x, so we can factor it out:

(1/2)^x+1 + (1/2)^x-2 + (1/2)^x+3 = (1/2)^x * ((1/2)^1 + (1/2)^(-2) + (1/2)^3) = (1/2)^x * (1/2 + 1/4 + 1/8).

Now, let's add the fractions inside the parentheses:

(1/2)^x * (1/2 + 1/4 + 1/8) = (1/2)^x * (4/8 + 2/8 + 1/8) = (1/2)^x * (7/8).

So now, our equation becomes:

(1/2)^x * (7/8) = 37.

To solve for x, we need to isolate (1/2)^x on one side of the equation. To do that, we'll multiply both sides by 8/7:

(8/7) * (1/2)^x * (7/8) = (8/7) * 37.

The (7/8) on the left side cancels out:

(8/7) * (1/2)^x = (8/7) * 37.

Now, we'll cancel out the 8/7 on both sides by multiplying each side by 7/8:

(7/8) * (8/7) * (1/2)^x = (7/8) * (8/7) * 37.

The (8/7) on the left side cancels out:

(1/2)^x = 37.

Finally, to solve for x, take the logarithm of both sides. The base of the logarithm doesn't matter, but let's use the natural logarithm (ln) for this example:

ln((1/2)^x) = ln(37).

Now, we can use the property of logarithms to bring down the exponent:

x * ln(1/2) = ln(37).

Now, solve for x:

x = ln(37) / ln(1/2).

Using a calculator:

x ≈ 5.209.

So, the solution to the equation is approximately x = 5.209.