Final answer:
Understanding the function f(x)=-ln(x-5) involves recognizing the inverse relationship between the exponential function and natural logarithm. The graph of this function will show a decrease, shifted 5 units to the right due to the horizontal shift and reflection over the x-axis. Examples of exponential growth and the properties of logarithms, such as expressing numbers in exponential form e.g., eln2, are related concepts.
Step-by-step explanation:
The question is about the function f(x)=-ln(x-5) and understanding its graph. Using properties of logarithms and exponential functions, one can analyze the behavior of this function. The inverse relationship between the exponential function and natural logarithm (ln) is key:
ln(ex) = x and elnx = x. To make sense of the function given, you need to know that ln(x) is defined for x>0, and typically the graph of ln(x) will start from the left and rise as it moves to the right. However, because our function is -ln(x-5), there is a horizontal shift to the right by 5 units and the negative sign indicates a reflection across the x-axis, which means it will be decreasing as it moves to the right.
For understanding the properties of growth and logarithmic functions, let's consider the example of compounding growth with the base e (Euler's number). If we wanted to express the number 2 in an exponential form, we could write it as eln2.
Regarding the graph of an exponential function, let's consider a general function f(x) = abx. The graph is typically a curve that changes differently based on the value of x. For example, if x is 0, the function simplifies to f(x) = ab0 = a, which represents the y-intercept.
In the case of the function f(x) = 0.25e-0.25x, when x = 0, we find that f(x) equals the constant 0.25, which is the maximum value this particular graph would have on the y-axis because the exponential portion equals 1.