For example, when I try to solve an equation like 1.096 = a^1000, what should I do?
I'm trying to find a formula for this set of data and that's how I got the above equation: d=???
Elevation (ft) Sea level labeled h
1000 2000 3000 4000
Takeoff roll (ft) labeled d
605 663 727 797 873
if you know about exponents, you should know about logs, which are the inverse function.
a^1000 = 1.096
1000 log(a) = log(1.096)
a = log(1.096)/1000 = 0.0000917
How did you get the 0.0000917?
I plugged log(1.096) divided by 1000 and got 3.98 times 10 to the -5.
For the data I gave above, we need to find a formula for the exponential function.
I got b to be 605. That part is right at least, right? Because of how 605 = b times a^0 which means b equals 605. (0 because the 605 matches up with sea level in the data--I posted it weirdly)
So then plugging b into the second set of data given, 663 = 605 times a^1000.
So 1.096 = a^1000
and then inserting what you taught me,
1000 log a = log 1.096
and so if a = 0.0000917, the formula of the function would be y=605 times 0.0000917^x?
We may have taken different logs. Base 10 vs base e.
Looks good to me. Does this fit any of the other data points very well?
To solve the equation 1.096 = a^1000, where you want to find the value of 'a,' you can follow these steps:
Step 1: Take the logarithm of both sides of the equation using a logarithmic function with base 'a.' This will help you remove the exponent and isolate 'a' on one side of the equation. Since the logarithm with base 'a' is not readily available, we will use the natural logarithm (ln) instead.
ln(1.096) = ln(a^1000)
Step 2: Apply the power rule of logarithms, which states that ln(a^b) = b * ln(a). Use this rule to simplify the equation.
1000 * ln(a) = ln(1.096)
Step 3: Divide both sides of the equation by 1000 to solve for ln(a).
ln(a) = ln(1.096) / 1000
Step 4: Use the exponential function to find the value of 'a' by applying the inverse of the natural logarithm.
a = e^(ln(1.096) / 1000)
Now, let's calculate the value of 'a' using a calculator:
a ≈ e^(ln(1.096) / 1000) ≈ 1.0009999
Therefore, the solution to the equation 1.096 = a^1000 is approximately a = 1.0009999.
Please note that due to limited decimal places in the approximation, the value may not be precisely 1.0009999, but it should be very close.