Solve for x: 9^x-3^x-8=0
since 9^x = (3^2)^x = (#^x(^2
we can let 3^x = y , then
9^x - 3^x - 8 = 0 becomes
y^2 - y - 8 = 0
y = (1 ± √33)/2
y = (1+√33)/2 or y = (1-√33)/2
3^x = (1+√33)/2
take log of both sides, and use log rules
x = log[(1+√33)/2] /log3 = appr 1.106
or
3^x = (1-√33)/2 which has no solution, since we will not be able to take the log of a negatiave number.
(3^2)^x-3^x-8=0,
(3^x)² -3^x -8 = 0,
3^x =y,
y² -y - 8=0,
y= (1±√33)/2.
Since 3^x>0, the negative root is nonsence,
=> y= (1+√33)/2 =3.37228,
3^x =3.37228,
x=log(3)3.37228= =ln3.37228/ln3≈1.21559/1.0986=1.1065
Test:
9^1.1065 – 3^1.1065 – 8 =
=11.372 – 3.372 – 8 = 0
To solve the equation 9^x - 3^x - 8 = 0 for x, we can use an algebraic method. However, in this case, it is not possible to solve it analytically using common algebraic techniques.
We can instead use a numerical method called approximation or iteration to find an approximate solution. One such method is the Newton-Raphson method.
1. Start with an initial guess for x. Let's assume x = 1.
2. Calculate the function value f(x) = 9^x - 3^x - 8 using the initial guess. In this case, f(1) = 9^1 - 3^1 - 8 = -2.
3. To approximate the slope of the function at that point, calculate the derivative f'(x) = d/dx (9^x - 3^x - 8). The derivative of 9^x is (9^x) * ln(9), and the derivative of 3^x is (3^x) * ln(3). Therefore, f'(x) = (9^x) * ln(9) - (3^x) * ln(3).
4. Calculate the next approximation of x by using the formula: x1 = x - f(x)/f'(x). In this case, x1 = 1 - (-2)/[(9^1) * ln(9) - (3^1) * ln(3)].
5. Repeat steps 2 to 4 using the new x value obtained in step 4 until you reach an acceptable level of accuracy or convergence.
6. Continue the process until you find a value of x for which f(x) is close to zero.
By continuously iterating through steps 2 to 4, you can find an approximate solution for x. Keep in mind that manually calculating these steps can be time-consuming. It may be more straightforward to use mathematical software or calculators that have built-in functions for numerical approximation methods.