// Numbas version: exam_results_page_options {"name": "Jamie's copy of Graphing logarithms of the form y=log_{b}(x) with b>1", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "

The following questions will gauge your understanding of logarithms and how to graph them.

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The logarithm you will be working with for this question is \\[y=\\log_{\\var{b}}(x).\\]

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The easiest type of exponential to graph where the base is greater than 1 and no transformations take place.

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The \$x\$-intercept of \$y=\\log_{\\var{b}}(x)\$ is the point \$\\large(\$[[0]], 0\$\\large)\$.

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Another easily found point on the curve is \$\\large(\$[[0]], \$\\large1 )\$.

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Given \$y=\\log_{\\var{b}}(x)\$, everytime \$x\$ increases by a factor of \$\\var{b}\$, \$y\$  [[0]].

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increases by 1.

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decreases by 1.

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increases by {b-1}.

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decreases by {b-1}.

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increases by a factor of {b}.

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decreases by a factor of {b}.

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As \$x\$ increases without bound, [[0]]

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\$y\$ approaches \$0\$.

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\$y\$ decreases without bound.

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\$y\$ increases without bound.

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\$y\$ approaches \$\\var{b}\$.

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Which graph best represents \$y=\\log_{\\var{b}}(x)\$?

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{graph1(1)}

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{graph1(2)}

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{graph1(3)}

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{graph1(4)}

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This question assumes you understand the definition and the laws of logarithms.

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a) To find the \$x\$-intercept, substitute \$y=0\$ into the equation: \$0=\\log_{\\var{b}}(x)\$ which is equivalent to \$\\var{b}^0=x\$, i.e. \$x=1\$. Therefore, the \$x\$-intercept is the point \$(1,0)\$.

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b) Substitute \$y=1\$ into the equation: \$1=\\log_{\\var{b}}(x)\$ which is equivalent to \$\\var{b}^1=x\$, i.e. \$x=\\var{b}\$. Therefore, another easily found point is \$(\\var{b},1)\$.

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c) Let's investigate what happens to the value of \$y\$ when the value of \$x\$ is increased by a factor of \$\\var{b}\$:

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\\[\\log_{\\var{b}}(\\var{b}x)=\\log_{\\var{b}}(\\var{b})+\\log_{\\var{b}}(x)=1+\\log_{\\var{b}}(x)\\]

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This is the old \$y\$ value plus 1, so we can say that \$y\$ is increased by 1.

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d) As \$x\$ gets larger and larger, \$y\$ gets larger and larger. Even though the rate of increase slows, \$y\$ continues to grow without bound.

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e) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve \$y=\\log_{\\var{b}}(x)\$ the closer \$x\$ gets to zero (approaching from the right), the smaller \$y\$ gets (without bound). In other words, as \$x\$ approaches \$0\$ from the right, \$y\$ approaches negative infinity. This means that the asymptote for \$y=\\log_{\\var{b}}(x)\$ is the line \$x=0\$ (the \$y\$-axis).

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f) Given all the information above, it should be clear that the graph should look like

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{graph1(1)}

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