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Problem 33 Hard Difficulty

(a) How is the logarithmic function $ y = \log_b x $ defined?
(b) What is the domain of this function?
(c) What is the range of this function?
(d) Sketch the general shape of the graph of the function $ y = \log_b x $ if $ b > 1 $.

Answer

a) $\log _{b} x=y \leftrightarrow b^{y}=x$
b) $(0, \infty)$
c) (c) $\mathbb{R}$
(d) See Figure 11

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Video Transcript

in this problem. We're looking at some of the most important features of the basic log rhythmic function Y equals log base be of X. And did you know that this came from trying to find the inverse of y equals B to the X power? So this is an exponential function y equals B to the X. If we found its inverse, it would be X equals B to the why so the definition of the log rhythmic function why equals log Base P of X is that it is equivalent to X equals B to the why now the base has to be a number that's greater than zero, and all their basic rules for exponential functions apply. All right, So what is the domain and what is the range of this log rhythmic function? Well, knowing that it's the inverse of the exponential function, how about if we start by finding the domain and range of the exponential function, and then we switch them to get the domain a range of the log rhythmic function. So we know the exponential function, particularly if the base is greater than one is going to look like this. It's domain is all real numbers and its range is. Why is greater than zero if we write those in interval notation we have the domain is negative. Infinity to infinity on the range is zero to infinity. So for the log rhythmic function, we're going to switch those and the domain will be what the range waas for the exponential and vice versa. So the domain will be zero to infinity. On the range will be negative. Infinity to infinity. Now what does the graph look like? Particularly if the base is greater than one. So it's going to look like the inverse of the exponential curve, which means the reflection of that curve across the line y equals X, and that's going to be something like this.