Questions: Solve the equation: 3^x=47 a. Give the exact answer. x=ln(47)/ln(3) b. Give the decimal approximation rounded to 4 dec . places. x=1

Transcript text: Solve the equation: $3^{x}=47$ a. Give the exact answer. $x=\frac{\ln (47)}{\ln (3)}$ b. Give the decimal approximation rounded to 4 dec . places. $x=1$

Solution

Solution Steps

To solve the equation $3^x = 47$, we need to use logarithms. The exact solution can be found by taking the natural logarithm of both sides, which allows us to solve for $x$. The decimal approximation can then be calculated using Python.

Step 1: Solve for $ x $ Using Logarithms

To solve the equation $ 3^x = 47 $, we take the natural logarithm of both sides:

\[ \ln(3^x) = \ln(47) \]

Using the property of logarithms that allows us to bring the exponent down, we have:

\[ x \cdot \ln(3) = \ln(47) \]

Step 2: Isolate $ x $

Next, we isolate $ x $ by dividing both sides by $ \ln(3) $:

\[ x = \frac{\ln(47)}{\ln(3)} \]

Step 3: Calculate the Exact Value

Calculating the exact value gives us:

\[ x \approx 3.504555375379736 \]

Step 4: Round to Four Decimal Places