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Fill in the blank.
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All the curves of logarithmic functions π of π₯ equals log base π of π₯, for any positive base π not equal to one, pass through the point what.
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To answer this question, it might be useful to begin by identifying what we understand about this logarithmic function.
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Suppose we rewrite our logarithmic function as an equation π equals log base π of π₯.
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This is equivalent to saying π to the πth power is equal to π₯.
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And so, with this in mind, we might be able to identify the point through which any function of this form must pass.
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And this comes from knowing some of our rules when it comes to dealing with exponents.
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Letβs suppose that exponent π is equal to zero; then no matter the value of π, π₯ will always be π to the zeroth power.
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But of course, weβre told that π is positive, so itβs not equal to zero.
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And itβs greater than zero, and itβs not equal to one.
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Now, we know if we raise a number of this form to the power of zero, we actually get one.
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So given the criteria for π, no matter its value in this case, when we raise it to zero, we get one.
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This corresponds to the point with coordinates one, zero on the graph of our function.
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And so all the curves of logarithmic functions of this form must pass through the point one, zero.
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Now, we can think about this in an alternative way.
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Letβs think about the graph of an exponential function π¦ equals π to the power of π₯.
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These graphs always pass through the point zero, one.
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Then we know that the logarithmic function is the inverse of the exponential function.
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This means we can map from the graph of the exponential function onto the graph of the logarithmic function by reflection across the line π¦ equals π₯.
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In this reflection, the point zero, one maps onto the point one, zero.
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So since all graphs of the form π¦ equals π to the power of π₯ pass through the point zero, one, all graphs of the form π¦ equals log base π of π₯ pass through the point one, zero.