The easiest type of exponential to
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=\\log_{\\var{b}}(x).\\]
\n", "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: $0=\\log_{\\var{b}}(x)$ which is equivalent to $\\var{b}^0=x$, i.e. $x=1$. Therefore, the $x$-intercept is the point $(1,0)$.
\nb) 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 $(\\var{b},1)$.
\nc) Let's investigate what happens to the value of $y$ when the value of $x$ is increased by a factor of $\\var{b}$:
\n\\[\\log_{\\var{b}}(\\var{b}x)=\\log_{\\var{b}}(\\var{b})+\\log_{\\var{b}}(x)=1+\\log_{\\var{b}}(x)\\]
\nThis is the old $y$ value plus 1, so we can say that $y$ is increased by 1.
\nd) As $x$ gets larger and larger, $y$ gets larger and larger. Even though the rate of increase slows, $y$ continues to grow without bound.
\ne) 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).
\nf) Given all the information above, it should be clear that the graph should look like
\n{graph1(1)}
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