Final answer:
To find the equation of a transformed logarithm that fits two given points, one sets up equations corresponding to each point and solves for constants in the logarithmic function. Utilizing logarithmic properties will help determine the values of these constants. Lastly, the base of the logarithm is adjusted to fit the data points.
Step-by-step explanation:
To write an equation for the transformed logarithm that passes through the points (2,0) and (-3,3), we need to find the equation of the form y = a · logb(x - h) + k that fits those points. The points give us two equations based on the general logarithmic function. For the point (2,0), the equation becomes 0 = a · logb(2 - h) + k, and for the point (-3,3), it becomes 3 = a · logb(-3 - h) + k.
First, we need to set up these equations to solve for a, b, h, and k. Since the point (2,0) tells us that y is zero, the whole right side of the equation must be zero, which suggests k must equal zero as logarithms do not naturally give a zero result without external transformations. By plugging in the values from the second point into the function, we can calculate the values for a and h using properties of logarithms such as the logarithm of a number that results from the division of two numbers being the difference between the logarithms of the original numbers (log(a/b) = log(a) - log(b)), and the logarithm of a number raised to an exponent being the product of the exponent and the logarithm of the number (log(an) = n · log(a)). Finally, we can fine-tune the base b to match the curves through both points.