lnx^10+ lnx^9 + …….lnx=10+9+8+…..1

To solve this equation, we need to understand the properties of logarithms and apply some algebraic manipulation.

First, let's rewrite the left side of the equation using the properties of logarithms. The sum of logarithms can be expressed as the logarithm of the product:

ln(x^10) + ln(x^9) + ... + ln(x) = ln(x^10 * x^9 * ... * x^1)

Next, simplify the product of the terms inside the logarithm:

ln(x^10 * x^9 * ... * x^1) = ln(x^(10 + 9 + ... + 1))

Now, we have the equation:

ln(x^(10 + 9 + ... + 1)) = 10 + 9 + ... + 1

The sum of numbers from 1 to n can be calculated using the formula for the sum of an arithmetic series:

Sum = (n/2) * (first_term + last_term)

In our case, the first term is 1 and the last term is 10. Using the formula, we get:

Sum = (10/2) * (1 + 10) = 55

Therefore, we can simplify the equation to:

ln(x^55) = 55

Now, to solve for x, we need to inverse the logarithm, taking the exponential of both sides:

e^(ln(x^55)) = e^55

Since the exponential function and the natural logarithm function are inverses of each other, we can simplify further:

x^55 = e^55

Finally, we take the 55th root of both sides to isolate x:

x = (e^55)^(1/55)

Thus, the value of x can be calculated by raising the constant e (approximately 2.71828) to the power of 55 and then taking the 55th root of that value.