Questions: Let's solve the exponential equation 3e^x=60. (a) First, we isolate e^x to get the equivalent equation . (b) Next, we take ln of each side to get the equivalent equation . (c) Now we use a calculator to find x= (Round your answer to three decimal places.)

Let's solve the exponential equation 3e^x=60.
(a) First, we isolate e^x to get the equivalent equation .
(b) Next, we take ln of each side to get the equivalent equation .
(c) Now we use a calculator to find x= (Round your answer to three decimal places.)

Transcript text: Let's solve the exponential equation $3 e^{x}=60$. (a) First, we isolate $e^{x}$ to get the equivalent equation $\square$ . (b) Next, we take In of each side to get the equivalent equation $\square$ . (c) Now we use a calculator to find $x=$ $\square$ (Round your answer to three decimal places.)

Solution

Solution Steps

To solve the exponential equation $3 e^{x} = 60$:

(a) First, we isolate $e^{x}$ by dividing both sides by 3. (b) Next, we take the natural logarithm (ln) of each side to solve for $x$. (c) Finally, we use a calculator to find the value of $x$ and round it to three decimal places.

Step 1: Isolate $ e^{x} $

Starting with the equation: \[ 3 e^{x} = 60 \] we divide both sides by 3 to isolate $ e^{x} $: \[ e^{x} = \frac{60}{3} = 20 \]

Step 2: Take the Natural Logarithm

Next, we take the natural logarithm of both sides: \[ \ln(e^{x}) = \ln(20) \] Using the property of logarithms, this simplifies to: \[ x = \ln(20) \]

Step 3: Calculate and Round $ x $

Calculating $ \ln(20) $ gives us approximately: \[ x \approx 2.995732273553991 \] Rounding this to three significant figures, we find: \[ x \approx 2.996 \]

Final Answer

\[ \boxed{x = 2.996} \]

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