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Solve ln 2 + ln x = 5
. Round to the nearest thousandth, if necessary.
To solve the equation ln 2 + ln x = 5, we can use the property of logarithms that states:
ln A + ln B = ln(AB)
Applying this property to the equation, we can rewrite it as:
ln (2x) = 5
Next, we need to eliminate the natural logarithm by exponentiating both sides of the equation. Exponentiating with base e (Euler's number) will cancel out the natural logarithm:
e^(ln(2x)) = e^5
Simplifying, we have:
2x = e^5
To solve for x, we substitute the value of e (approximately 2.71828) into the equation:
2x ≈ 2.71828^5
Using a calculator, we find:
2x ≈ 148.41316
To isolate x, we divide both sides of the equation by 2:
x ≈ 148.41316 / 2
Simplifying, we get:
x ≈ 74.20658
Therefore, the solution to ln 2 + ln x = 5, rounded to the nearest thousandth, is x ≈ 74.207.