Solve the exponential equation. Give the exact value for x. (Express all logarithmic functions in terms of ln(x) in your answer.)
text((a) ) e^x = 10
x =
text((b) ) e^x = 2.0
x =
text((c) ) e^x = 37.9
x =
To solve exponential equations like these, we need to take the natural logarithm (ln) of both sides. By doing this, we can eliminate the exponential term and solve for x.
a) To solve the equation e^x = 10:
Take the natural logarithm (ln) of both sides:
ln(e^x) = ln(10)
Using the property of logarithms that ln(e^x) = x:
x = ln(10)
b) To solve the equation e^x = 2.0:
Similarly, take the natural logarithm of both sides:
ln(e^x) = ln(2.0)
x = ln(2.0)
c) To solve the equation e^x = 37.9:
Again, take the natural logarithm of both sides:
ln(e^x) = ln(37.9)
x = ln(37.9)
In all three cases, the solution for x is the natural logarithm of the constant on the right side of the equation.