I am trying to calculate the value of $\log_b(x)$ for any $b\in(0, \infty)$ and any $x>0.$ I am supposed to do this only using basic arithmetic operations and exponentiation.
I know that in order to find zeros using the secant method, you use the following formula.
$$x_{k+1} = x_k - f(x_k)\frac{x_k - x_{k-1}}{f(x_k) - f(x_{k-1})}$$
But I have no idea what $f(x)$ will be in this case. Any pointers to get me on the right track?
Thank you.
Note that $b\ne 1$. Also, since you are finding a root, and you want this to be the logarithm, I'm going to use the following notation $$x=\log_b y$$ Then from the definition of the logarithm, $$b^x=y$$ You can rewrite this as $$b^x-y=0$$ So finding $\log_b y$ is the equivalent of finding the root of the $f(x)=0$ equation, where $$f(x)=b^x-y$$ A good starting point is $x_0=0$.
Can you take it from here?