Question

4^(х-1) + 4^х = 64/5

Answer 1

Ответ: x = 4 - 2log₄5.

Пошаговое объяснение:

4^х/4 + 4^х = 64/5,

(4^х+ 4*4^x)/4 = 64/5,

(4^x(1 + 4))/4 = 64/5,

5*4^x/4 = 64/5  умножим обе части на (4/5),

4^x = (64*4)/(5*5) = 4^4/5^2.

Прологарифмируем обе части по основанию 4.

х = 4 - 2log₄5 ≈ 1,67807.

Answer 2

To solve the equation 4^(x-1) + 4^x = 64/5, we can first combine the terms with the same base (4) and then proceed to solve for 'x'.

Step 1: Combine the terms with the same base (4). 4^(x-1) + 4^x = (4^x * 4^(-1)) + 4^x = 4^x * (1/4) + 4^x = (1/4) * 4^x + 4^x = (1/4 + 1) * 4^x = (5/4) * 4^x

Step 2: Rewrite 64/5 as a fraction with a common denominator. 64/5 = (64 * 4) / (5 * 4) = 256/20

Now, the equation becomes: (5/4) * 4^x = 256/20

Step 3: Simplify the equation further. To remove the fraction, multiply both sides by (4/5): (5/4) * (4/5) * 4^x = (256/20) * (4/5) 4^x = 1024/100

Step 4: Write both sides of the equation with the same base (4). 4^x = (4^5) / (4^2) 4^x = 4^(5-2)

Step 5: Set the exponents equal to each other. x = 5 - 2 x = 3

Therefore, the solution to the equation is x = 3. You can verify this by substituting x = 3 back into the original equation to ensure it holds true.