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

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.

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:

$$x \cdot \ln(3) = \ln(47)$$

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

$$x = \frac{\ln(47)}{\ln(3)}$$

Calculating the exact value gives us:

$$x \approx 3.504555375379736$$

Rounding this value to four decimal places, we find:

$$x \approx 3.5046$$

Final Answer