# Logarithmic Function Calculator

Instructions: Use this step-by-step Logarithmic Function Calculator, to find the logarithmic function that passes through two given points in the plane XY. You need to provide the points $$(t_1, y_1)$$ and $$(t_2, y_2)$$, and this calculator will estimate the appropriate exponential function and will provide its graph. First t ($$t_1$$) = First y ($$f(t_1)$$) = Second t ($$t_2$$) = Second y: ($$f(t_2)$$) = Points to evaluate (Optional. Comma or space separated) =

## Logarithmic Function Calculator from Two Points

The main purpose of this calculator is to estimate the parameters $$A_0$$ and $$k$$ for the logarithmic function $$f(t)$$ which is defined as:

$f(t) = A_0 \ln(k t)$

The parameters need to be so the logarithmic function passes through the two given points $$(t_1, y_1)$$ and $$(t_2, y_2)$$.

### How do you estimate a logarithmic function from two points?

Algebraically speaking, you need to solve the following system of equations to find the parameters $$A_0$$ and $$k$$:

$y_1 = A_0 \ln(k t_1)$ $y_2 = A_0 \ln(k t_2)$

By solving this system for the unknowns $$A_0$$ and $$k$$, we can find unique solutions, as long as $$t_1 = \not t_2$$.

Indeed, by subtracting both sides of the equations:

$\displaystyle y_1 - y_2 = A_0 \left( \ln(k t_1) - \ln(k t_2) \right)$ $\displaystyle \Rightarrow \, y_1 - y_2 = A_0 \ln \left(\displaystyle\frac{k t_1}{k t_2}\right)$ $\displaystyle \Rightarrow \, y_1 - y_2 = A_0 \ln \left(\displaystyle\frac{t_1}{t_2}\right)$ $\Rightarrow \, A_0 = \displaystyle \frac{y_1 - y_2}{\ln(t_1) - \ln(t_2)}$

which solves the equations for $$A_0$$. Now, in order to solve for $$k$$ we use the first equation and apply exponential to both sides::

$y_1 = A_0 \ln(k t_1)$ $\Rightarrow \, \displaystyle e^{\frac{y_1}{A_0}} = k t_1$ $\Rightarrow \, k = \displaystyle \frac{e^{\frac{y_1}{A_0}}}{t_1}$

and there we have found $$k$$, as function of $$A_0$$ that is already determined and known.

### How do you calculate an exponential function?

If instead of a logarithmic function you are interested in exponential behavior, then you should probably use this exponential function calculator , which follows the same logic of estimating parameters to enforce the function passing through two given points.