WEBVTT
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Let's talk about logarithmic functions.
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The logarithmic function with base a, for a
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is greater than 0 and not equal to one, is
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defined by the following: y equals log base
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a of x if and only if x equals a to the y.
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Now remember we said that a needs to be
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greater than zero, a should not equal 1, and
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there's one more restriction we should discuss.
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Notice that the base a is a positive real
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number not equal to 1. But that means here
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that if we raise a positive real number to
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another real number power, the result will
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always be a positive number.
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Therefore, x needs to be greater than zero.
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Now this is probably the most important
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thing about logarithmic functions, so let's
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put a box around it and remember it.
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Now let's practice using this.
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Instructions are to rewrite in exponential form.
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Here's our first example: log base 3 of k equals p.
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Now let's see. The first thing I want to pick out is the
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base of my logarithm. The base is 3.
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That should be the base of the exponential form.
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Now I need to find an exponent. Well, the exponent is y. Notice here that
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y is equal to the logarithm. So, that means that a logarithm is an exponent.
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Let's look down here. What's equal to our logarithm?
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p, so that p is my exponent. Alright, what's left?
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x, x is the argument of the logarithm. The argument here is k.
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Alright, so log base 3 of k equals p is
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equivalent to 3 to the p equals k. Let's try another example.
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This logarithm looks kind of funny right. This is a natural log.
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Alright, the natural log is the base e
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logarithm. So our base is e, and I'm going to look for my exponent.
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The exponent should be equal to the
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logarithm. The logarithm is an exponent so 52. What's left?
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x, the argument, and the argument here happens to be x.
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So, this natural log of x equals 52 is equivalent
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to e to the fifty-second power equals x.
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Let's look at a couple more examples.
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For our next two examples, let's look at rewriting in logarithmic form.
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Example 3: again the first thing we want to pick out is the base.
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The base of this exponential function here
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is 10, so that'll be the base of a logarithm.
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Base 10 is another special log, this one is
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the common log, and it is special we just don't write the base
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because it's understood to be 10.
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So, we're going to have log base 10, I'd
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like to leave a space to fill in later
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because the easiest thing to pick out now would be the exponent.
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The exponent here is x, and the exponent is equal to the logarithm.
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So, the x should be here. x is the
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logarithm, and that leaves me with the 15, and the 15 must be the argument.
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So, 15 equals 10 to the x is equivalent to
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the log of 15 equals x. Example 4: here we have x to
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the 2.2 power equals k. Again, identify the base of your logarithm.
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It's the same as the base of the exponential
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function here, so that would be x.
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Log base x of something has to be our
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exponent, and the exponent is 2.2.
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What's left? k, so k is the argument.
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x to the 2.2 power equals k is equivalent to log base x of k equals 2.2.
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Let's look at some other ways to apply the definition of a logarithm.
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For our next examples, the instructions ask
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us to find the exact value of a logarithmic
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expression without using the calculator.
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Okay, so let's look at the first one: log base 8 of 8.
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First let's write down what this is asking us to think about.
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The base is 8 and a logarithm is an
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exponent, so we're looking for the power of 8 where we get 8.
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Eight to what power is 8? Of course, it's the first, so this is 1.
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Let's try another one: log base 2 of 32.
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So we want to take our base, 2, and figure out
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what power do we have to raise 2 to, to get 32.
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Five, 2 to the fifth power is 32. Let's try some more.
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Example 7: log base 3 of 1/9 is what number?
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Again it's asking us to take the base 3 and
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determine the power we have to raise 3 to, to get 1/9.
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Okay, well look at the 1/9, we can rewrite
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that, that's 9 is 3 squared, so we can write 1 over 3 squared.
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We can also write that as 3 to the negative 2 power.
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Now it's clear that the exponent that I'm
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looking for must be negative 2. So, log base 3 of 1/9 is negative 2.
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Negative 2 is the exponent on 3 that will give you 1/9.
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Let's try number 8: this is that funny
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logarithm again, the natural log.
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Remember the base for natural log? Base e, right.
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Okay, so the base is e, we want to know the
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power we have to raise e to, to get the argument e to the third.
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So you can see that the power must be 3.
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Let's explore logarithmic functions further.
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Example 9: find the domain of the function y equals log base 2 of x minus 3.
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So we need to remember that in the definition of a logarithm,
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there's a restriction on x. x has to be greater than 0, again that's
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because with a base a that's greater than
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zero, when you raise that positive real
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number to another real number power, you'll always get a positive number.
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So again, what that means is the needs to be greater than 0.
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Here's the argument. Alright, let's look at our function.
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The argument here is x minus 3, so
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we need x minus 3 to be greater than 0.
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Add 3 to both sides, that tells us x needs to be greater than 3.
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We'd like to write this in interval notation.
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So, we want all the numbers bigger than 3,
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but we don't want to include 3 itself.
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Let's look at the graph of a logarithmic function.
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Example 10: I'd like to sketch the graph of
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the function y equals the natural log of x plus 2.
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I'd like to do this using transformations,
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so the first thing that I want to graph is
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the function y equals the natural log of x.
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I need to plot two points. Let's take x equal to 1.
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Then, the natural log of 1 is 0, so there's my
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first point, and for a second point,
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let's take x to be the base, e. The natural log of e is 1.
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Now I have two points to plot. Here's the point (1,0).
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Notice that's my x-intercept, and then the point (e,1).
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Remember e is about 2.7. Connect them in the general shape of a
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logarithmic graph noticing that right now
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the y-axis is the vertical asymptote.
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This equation is x equals 0, and that is the asymptote.
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Now let's look, we're doing transformations,
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so how do I transform the graph?
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The plus 2 is inside with the argument, so
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which direction should we move?
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Correct, we want to go to the left 2 units.
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Alright, now remember that asymptote I pointed out.
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We need to move that first. We need to go to the left 2 units.
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We're going to draw a dotted line for the asymptote.
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This is now the vertical line x equals negative 2,
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now we can move the points on the graph to the left 2 units.
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So, one, two. This one is about 2.7, so when we move
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that 2 units to the left. It will be about 0.7.
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So, here's the graph of the natural log of x plus 2.
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We've already pointed out one important
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feature, and that is the vertical asymptote: x equals negative 2.
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Another important characteristic would be
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the domain of the function. Notice that x has to be something bigger
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than negative 2, it can't be to the left of that,
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and it can't be equal to negative 2,
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so anything to the right. x must be greater than negative 2.
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We want that in interval notation: that
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would be the interval from negative 2 to infinity.
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Remember the negative 2 is not included, and then finally the range.
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y can be any real number you can imagine,
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so the range is going to be the set
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of all real numbers and in interval notation, negative infinity to infinity.
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Continue to practice using the definition of
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a logarithm and working with graphs of
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logarithms until you build skill and understanding of the concepts.