Solve for t using natural logarithms.
7e^3t = 10e^2t
To solve for t in the equation 7e^(3t) = 10e^(2t), we can use natural logarithms (ln). Here's the step-by-step process:
Step 1: Take the natural logarithm of both sides of the equation.
ln(7e^(3t)) = ln(10e^(2t))
Step 2: Apply the logarithmic property ln(ab) = ln(a) + ln(b) to simplify the equation.
ln(7) + ln(e^(3t)) = ln(10) + ln(e^(2t))
Step 3: Simplify the expressions with natural logarithm.
ln(7) + 3t = ln(10) + 2t
Step 4: Move the terms with t to one side of the equation.
3t - 2t = ln(10) - ln(7)
Step 5: Combine like terms.
t = ln(10) - ln(7)
Step 6: Use a calculator to find the decimal approximation of ln(10) - ln(7).
t ≈ 0.3567
Therefore, the value of t, using natural logarithms, is approximately 0.3567.