Can you show me how to solve this using logarithims? thank you
3e^-3x = 10.21 for x.
Sure! To solve the equation 3e^(-3x) = 10.21 for x using logarithms, we can take the natural logarithm (ln) of both sides of the equation. Here's how to do it step by step:
Step 1: Start with the equation 3e^(-3x) = 10.21
Step 2: Take the natural logarithm (ln) of both sides of the equation:
ln(3e^(-3x)) = ln(10.21)
Step 3: Use the logarithmic property ln(ab) = lna + lnb to simplify the equation:
ln(3) + ln(e^(-3x)) = ln(10.21)
Step 4: Use another property of logarithms, ln(e^(-3x)) = -3x, since ln(e) = 1:
ln(3) - 3x = ln(10.21)
Step 5: Move the -3x term to the other side of the equation:
-3x = ln(10.21) - ln(3)
Step 6: Divide both sides of the equation by -3 to isolate x:
x = (ln(3) - ln(10.21))/3
Step 7: Use a calculator to calculate the value of x:
x ≈ -0.145
So, the solution to the equation 3e^(-3x) = 10.21 for x, using logarithms, is approximately x = -0.145.