Questions: Solve the equation. 45 e^(7 k)=225 If 45 e^(7 k)=225, then k=

Transcript text: Solve the equation. \[ 45 e^{7 k}=225 \] If $45 e^{7 k}=225$, then $k=$ $\square$ (Type an exact answer.)

Solution

Solution Steps

To solve the equation $45 e^{7k} = 225$, we first isolate the exponential term by dividing both sides by 45. Then, we take the natural logarithm of both sides to solve for $k$.

Step 1: Isolate the Exponential Term

Given the equation $45 e^{7k} = 225$, we first divide both sides by 45 to isolate the exponential term: \[ e^{7k} = \frac{225}{45} = 5 \]

Step 2: Take the Natural Logarithm

To solve for $k$, we take the natural logarithm of both sides: \[ \ln(e^{7k}) = \ln(5) \] Using the property $\ln(e^x) = x$, we have: \[ 7k = \ln(5) \]

Step 3: Solve for $k$

Divide both sides by 7 to solve for $k$: \[ k = \frac{\ln(5)}{7} \]