Find the exponential function f(x)=a⋅2^bx with (0,3) and (1,4). a=?,b=?
plug in your values, assuming you meant
f(x) =a (2)^(bx) and not what you typed.
for (0,3) , 3 = a(2)^0 --> 3 = a
for (1,4) , 4 = 3(2)^b
2^b = 4/3
take logs of both sides, and use log rules
b log2 = log4 - log3
b = (log4 - log3)/log2 = appr .415037
thank you!
Oh, the joy of exponential functions! Let me put on my clown shoes to solve this puzzle for you.
First, let's plug in the point (0,3) into our equation:
3 = a⋅2^0
Well, anything raised to the power of 0 is always 1, so we can simplify:
3 = a(1)
3 = a
So, we found that a = 3. One down, one to go!
Now let's plug in the point (1,4) into our equation:
4 = 3⋅2^b(1)
Simplifying further:
4 = 3⋅2^b
Now Mr. Clown needs to use his clown math skills to solve for b. Let's divide both sides by 3:
4/3 = 2^b
To get rid of that pesky exponent, let's take the logarithm of both sides. But wait, clowns don't really like logarithms, so we'll be doing some guesswork:
Taking a wild guess, let's use base 2 logarithm, or log2:
log2(4/3) = b
Approximating this, we find:
b ≈ 0.4150375
Voila! We found our values. So the exponential function f(x) = 3⋅2^(0.4150375x), with a ≈ 3 and b ≈ 0.4150375.
Please take my answer with a grain of confetti, as some numerical approximation was involved. Have a clown-tastic day! 🎪
To find the values of a and b in the exponential function f(x) = a * 2^(bx), you can use the two given points (0,3) and (1,4) to create a system of equations.
Step 1: Plug in the first point (0, 3) into the function f(x) = a * 2^(bx).
3 = a * 2^(0 * b)
3 = a * 2^0
3 = a * 1
a = 3
Step 2: Plug in the second point (1, 4) into the function f(x) = a * 2^(bx).
4 = 3 * 2^(1 * b)
4 = 3 * 2^b
We now have a system of equations:
a = 3
4 = 3 * 2^b
Step 3: Solve the second equation for b.
Divide both sides by 3:
4/3 = 2^b
Step 4: Take the logarithm (base 2) of both sides to solve for b.
log2(4/3) = log2(2^b)
log2(4/3) = b
Therefore, we have determined that a = 3 and b = log2(4/3).
Note: Depending on your calculator or software, you may need to convert the answer in terms of natural logarithm (ln) instead of base 2 logarithm. In that case, you would use ln(4/3)/ln(2) to find b.