Solve the equation. Check for extraneous solutions.
10 ln 100x - 3 = 117
Thank you in advanced
To solve the equation 10 ln(100x) - 3 = 117, we can start by isolating the natural logarithm term.
1. Add 3 to both sides of the equation:
10 ln(100x) = 120
2. Divide both sides by 10:
ln(100x) = 12
Next, we can exponentiate both sides using the property of logarithms. Since the natural logarithm is the inverse of the exponential function, we can write the equation as:
3. Rewrite the equation in exponential form:
e^(ln(100x)) = e^(12)
100x = e^(12)
Now, we can solve for x by dividing both sides by 100:
4. Divide both sides by 100:
x = e^(12) / 100
To check for extraneous solutions, we need to ensure that the value of x we obtain is valid for the original equation. ln(100x) is only defined for positive values of 100x.
Since e^(12) is a positive value, we can conclude that x = e^(12) / 100 is a valid solution without extraneous solutions.
Therefore, the solution to the equation 10 ln(100x) - 3 = 117 is:
x = e^(12) / 100
To solve the equation 10 ln(100x) - 3 = 117, we need to isolate the variable x.
1. Start by adding 3 to both sides of the equation:
10 ln(100x) = 120.
2. Divide both sides of the equation by 10:
ln(100x) = 12.
3. Now, we need to eliminate the natural logarithm (ln) by exponentiating both sides of the equation with the base e (the Euler's number):
e^(ln(100x)) = e^12.
By the properties of logarithms and exponents, the natural logarithm and the exponential function cancel each other out, leaving us with:
100x = e^12.
4. Finally, divide both sides of the equation by 100:
x = e^12 / 100.
To check for extraneous solutions, substitute the value of x back into the original equation and see if it satisfies the equation.
Therefore, the solution to the equation is x = e^12 / 100.
10 log(100 x)-3 = 117
Add 3 to both sides:
10 log(100 x)- 3 = 117 + 3
10 log(100 x) = 120
Divide both sides by 10:
log(100 x) = 12
Cancel logarithms by taking exp of both sides:
100 x = e^12
Divide both sides by 100:
x = e^12/100
Now test that this solution is appropriate by substitution into the original equation:
Check the solution x = e^12/100:
10 log(100 x)- 3 = -3 + 10 log((100 e^12)/100) = 117
So the solution is correct.
Thus, the solution is:
x = e^12/100