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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.

user20057611
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1 Answers1

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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?

Andrei
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