Question

What is the true solution to ln 20+ln 5= 2 ln x

Answer 1

The true solution to the equation ln 20 + ln 5 = 2 ln x is x = 10. This is found by using properties of logarithms, namely that ln(xy) = ln x + ln y, followed by equating the exponents and taking the square root.

Step-by-step explanation:

The student asked for the solution to the equation ln 20 + ln 5 = 2 ln x. To solve this, we can use a property of logarithms, specifically that the logarithm of a product of two numbers is the sum of the logarithms of those numbers: ln(xy) = ln x + ln y. Applying this to the equation gives us ln(20 × 5) = 2 ln x or ln 100 = 2 ln x. This simplifies to ln 100 = ln(x²), and since the natural logarithm is an inverse function to the exponential function, we can equate the arguments of the ln resulting in 100 = x². To find x, we can take the square root of both sides, so x = ±10. Since x represents a length and cannot be negative, we discard the negative solution, leaving us with x = 10.

Answer 2

The true solution to the
`\ln 20 + \ln 5 = 2 \ln x` is, 10

Step-by-step explanation:

Given the equation:
`\ln 20 + \ln 5 = 2 \ln x`

Using logarithmic rules:

• 
  `\ln(mn) = \ln m+ \ln n`

• 
  `\ln a^b = b \ln a`

• 
  `ln a = \ln b` ⇒
  `a = b`

Now, using these properties solve for x;

`\ln (20 \cdot 5) = 2\ln x`

`\ln (100) = \ln x^2`

`\ln 10^2 = \ln x^2`

On comparing both sides we have;

`x^2 = 10^2`

or

`x = √(10^2)`

x = 10

Therefore, the true solution to the given equation is, x= 10