Graphing $y=a\\log_{b}(x)+c$
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", "name": "a"}}, "statement": "The following questions will gauge your understanding of logarithms and how to graph them.
\nThe logarithm you will be working with for this question is \\[y=\\simplify{{a}log(x,{b})+{c}}.\\]
\n\n", "extensions": ["jsxgraph"], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "This question assumes you understand the definition and the laws of logarithms.
\na) To find the $x$-intercept, substitute $y=0$ into the equation and solve for $x$:
\n$\\begin{align*}&&0&=\\simplify{{a}log(x,{b})+{c}}\\\\\\implies&& \\var{-c}&=\\simplify{{a}log(x,{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{-{c/a}}&=\\simplify{log(x,{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{{b}^(-{c/a})}&=x\\end{align*}$
\nTherefore, the $x$-intercept is the point $\\left(\\simplify[fractionNumbers,simplifyFractions,basic]{{b}^(-{c/a})},0\\right)$.
\nb) Substitute $x=\\var{b}$ into the equation: $y=\\simplify[!collectNumbers, basic]{{a}log({b},{b})+{c}}=\\simplify[!collectNumbers]{{a}*1+{c}}=\\var{a+c}$. Therefore, another easily found point is $(\\var{b},\\var{a+c})$.
\nc) As $x$ gets larger and larger (increases without bound, or approaches infinity) $\\simplify{{a}log(x,{b})+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).
\nd) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\\simplify{{a}log(x,{b})+{c}}$ the closer $x$ gets to zero (approaching from the right), the smaller larger $y$ 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).
\ne) Given all the information above, it should be clear that the graph should look like
\n{graph1(1)}
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\nNote: You should input the exact answer, not just a decimal approximation.
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