Certainly! It's great that Stella solved 3^(x)=9 by recognizing that 3^2 equals 9, so x must be 2. Now, for the equation 3^(x)=16, let's be a little more investigative since it's not as straightforward. Here we will use the "guess and check" method, which means we'll make educated guesses for the value of x and see how close the result is to 16.
Let's put on our detective hats!
Since we know that 3^2 is 9, and 3^3 is 27, the value of x we are looking for must be somewhere between 2 and 3 (because 16 is between 9 and 27).
Now, let's try a few guesses between 2 and 3 to see how close we can get to 16.
For x = 2.5,
3^2.5 ≈ 15.59.
Hmm, that’s pretty close to 16, but it’s a bit less.
Let's try a little higher value. How about x = 2.6?
3^2.6 ≈ 16.36.
This is a tad over 16, but even closer than before!
Let's get a little more granular. Try x = 2.55,
3^2.55 ≈ 15.97.
Bingo! We’re almost there. This is really close to 16.
With these educated guesses, we can deduce that the value of x for which 3^x is equal to 16 is approximately 2.55. You can use a calculator to refine further if you like, but 2.55 is accurate to two decimal places.
Using the "guess and check" method this way is similar to being a detective on a number hunt. You make educated guesses based on the information you have, and refine your guesses as you get closer to the solution. It’s an effective technique for solving equations like this without the use of logarithms, especially when you're trying to develop an intuitive understanding of how the values are related.
So, let's tell Stella that x is approximately 2.55, and congratulate her on being such a great numbers detective!