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The graph of every exponential function \(f( x) =a^{x}\), \(a>0\) and \(a\neq 1\), passes through three points: _____, _____, and _____.

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True or False  The graph of the exponential function \(f( x) =\left( \dfrac{3}{2}\!\right) ^{\!\!\!x}\) is decreasing.

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If \(3^{x}=3^{4}\), then \(x=\) _____.

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If \(4^{x}=8^{2}\) then \(x=\) _____.

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True or False  The graphs of \(y=3^{x}\) and \(y=\left( \dfrac{1}{3}\!\right) ^{\!\!\!x}\) are symmetric with respect to the line \(y=x\).

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True or False  The range of the exponential function \( f(x) = a^{x}\), \(a>0\) and \(a\neq 1\), is the set of all real numbers.

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The number \(e\) is defined as the base of the exponential function \(f\) whose tangent line to the graph of \(f\) at the point \( ( 0,1) \) has slope _____.

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The domain of the logarithmic function \(f( x) =\log _{a}x\) is _____.

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The graph of every logarithmic function \(f( x) =\log _{a}x\), \(a>0\) and \(a\neq 1\), passes through three points: _____, _____, and _____.

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Multiple Choice  The graph of \(f( x) =\log _{2}x\) is [(a) increasing, (b) decreasing, (c) neither].

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True or False  If \(y=\log _{a}x\), then \(y=a^{x}.\)

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True or False  The graph of \(f( x) =\log _{a}x\), \(a>0\) and \(a\neq 1\), has an \(x\)-intercept equal to \(1\) and no \(y\) -intercept.

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True or False  \(\ln e^{x}=x\) for all real numbers.

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\(\ln e\)= _____.

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Explain what the number \(e\) is.

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What is the \(x\)-intercept of the function \(h( x) =\ln ( x+1) ?\)

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Suppose that \(g( x) =4^{x}+2\).

1. What is \(g( -1) \)? What is the corresponding point on the graph of \(g\)?
2. If \(g( x) =66\), what is \(x\)? What is the corresponding point on the graph of \(g\)?

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Suppose that \(g( x) =5^{x}-3\).

1. What is \(g ( -1) \)? What is the corresponding point on the graph of \(g\)?
2. If \(g( x) =122\), what is \(x\)? What is the corresponding point on the graph of \(g\)?

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\(f( x) =2^{x+2}\)

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\(f( x) =1-2^{-x/3}\)

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\(f( x) =4\left( \dfrac{1}{3}\!\right) ^{\!\!\!x} \)

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\(f( x) =\left( \dfrac{1}{2}\!\right) ^{\!\!\!-x}+1\)

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\(f( x) =e^{-x}\)

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\(f( x) =5-e^{x}\)

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\(F( x) =\log _{2}x^{2}\)

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\(g( x) =8+5\ln ( 2x+3)\)

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\(f( x) =\ln ( x-1)\)

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\(g( x) = \sqrt{\ln x}\)

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\(f( x) =\ln ( x+4)\)

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\(f( x) =\dfrac{1}{2}\log ( 2x)\)

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\(f( x) =3e^{x}+2\)

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\(f( x) =2^{x/3}+4\)

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How does the transformation \(y=\ln ( x+c) \), \(c>0\), affect the \(x\)-intercept of the graph of the function \(f( x) =\ln x?\)

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How does the transformation \(y=e^{cx}\), \(c>0\), affect the \(y\)-intercept of the graph of the function \(f( x) =e^{x}?\)

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\(3^{x^{2}}=9^{x}\)

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\(5^{x^{2}+8}=125^{2x}\)

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\(e^{3x}=\dfrac{e^{2}}{e^{x}}\)

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\(e^{4x}\cdot e^{x^{2}}=e^{12}\)

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\(e^{1-2x}=4\)

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\(e^{1-x}=5\)

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\(5(2^{3x}) =9\)

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\(0.3(4^{0.2x}) =0.2\)

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\(3^{1-2x}=4^{x}\)

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\(2^{x+1}=5^{1-2x}\)

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\(\log _{2}(2x+1) =3\)

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\(\log _{3}( 3x-2) =2\)

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\(\log _{x}\left( \dfrac{1}{8}\!\right) =3\)

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\(\log _{x}64=-3\)

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\(\ln ( 2x+3) =2\ln 3\)

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\(\dfrac{1}{2}\log_{3}x=2\log _{3}2\)

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\(\log _{5}( x+1) -\log _{4}( x-2) =1\)

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\(\ln x=x\)

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\(e^{x}+\ln x=4\)

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\(e^{x}=x^{2}\)

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1. If \(f( x) =\ln ( x+4) \) and \(g( x) =\ln ( 3x+1) \), graph \( f \) and \(g\) on the same set of axes.
2. Find the point(s) of intersection of the graphs of \(f\) and \(g\) by solving \(f( x) =g( x)\) . Label any intersection points on the graph drawn in (a).
3. Based on the graph, solve \(f( x) > g( x).\)

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1. If \(f( x) =3^{x+1}\) and \(g( x) =2^{x+2}\), graph \(f\) and \(g\) on the same set of axes.
2. Find the point(s) of intersection of the graphs of \(f\) and \(g\) by solving \(f( x) =g( x) .\) Round answers to three decimal places. Label any intersection points on the graph drawn in (a).
3. Based on the graph, solve \(f( x) >g( x).\)