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Questions testing rather basic understanding of the index laws.

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(a) Add the powers: $\\var{m} + \\var{n}=\\var{m+n}$ (first index law).

(b) Multiply the powers $\\var{m1} \\times \\var{n1}=\\var{m1*n1}$ (second index law).

(c) Multiply $\\var{xx}$ and $\\var{y}$ to get $\\var{xx*y}$, with the power unchanged at $\\var{t}$. [This is the only way for the power to be a prime.]

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Express the following as a single power of $\\var{a}$:

\\[\\var{a}^{\\var{m}} \\times \\var{a}^{\\var{n}}\\]

Enter the power [[0]]

Input as a fraction or an integer, not as a decimal.

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Express the following as a single power of $\\var{b}$:

\\[(\\var{b}^{\\var{m1}})^{\\var{n1}}\\]

Enter the power [[0]]

Input as a fraction or an integer, not as a decimal.

Simplify as power of $\\var{x5}:$

${\\frac{1}{\\var{x5}^\\var{m52}}}$ = [[0]]

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Simplify as in the form $a \\sqrt{b}$ such that b is not a square number (use \"sqrt\" for square root):

$\\sqrt{\\simplify{({xx1}^2)*{kk1}}} $ = [[0]]

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Questions testing rather basic understanding of the index laws.

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Exponential functions $y = k a^x.$ Growth or decay, x-y intercepts.

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Identify if the graph $y=\\var{m} e^x$ is exponential growth or decay (or increasing decreasing function)

[[0]]

Does it have an x intercept

[[1]]

What is the y intercept

[[2]]

Use laws for addition and subtraction of logarithms to simplify a given logarithmic expression to an arbitrary base.

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Simplify the expressions to fill in the gaps.

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a)

We need to use the rule

\\[\\log_a(x)+\\log_a(y)=\\log_a(xy)\\text{.}\\]

Substituting in our values for $x$ and $y$ gives

\\[\\begin{align}
\\log_a(\\var{x1[1]})+\\log_a(\\var{x1[0]})&=\\log_a(\\var{x1[1]}\\times \\var{x1[0]})\\\\
&=\\log_a(\\var{x1[1]*x1[0]})\\text{.}
\\end{align}\\]

b)

We need to use the rule

\\[\\log_a(x)-\\log_a(y)=\\log_a\\left(\\frac{x}{y}\\right)\\text{.}\\]

Substituting in our values for $x$ and $y$ gives

\\[\\begin{align}
\\log_a(\\var{x1[4]*y1})-\\log_a(\\var{x1[4]})&=\\log_a(\\var{x1[4]*y1}\\div \\var{x1[4]})\\\\
&=\\log_a(\\var{y1})\\text{.}
\\end{align}\\]

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$\\log_a(\\var{x1[1]})+ \\log_a(\\var{x1[0]})=\\log_a($ [[0]]$)$

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When adding and subtracting logarithms we can simplify the expressions using some logarithm laws. These laws are

\\[\\begin{align}
\\log_a(x)+\\log_a(y)&=\\log_a(xy)\\text{,}\\\\
\\log_a(x)-\\log_a(y)&=\\log_a\\left(\\frac{x}{y}\\right)\\text{.}
\\end{align}\\]

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$\\log_a(\\var{(x1[4])*y1})-\\log_a(\\var{x1[4]})=\\log_a($ [[0]]$)$

When adding and subtracting logarithms we can simplify the expressions using some logarithm laws. These laws are

\\[\\begin{align}
\\log_a(x)+\\log_a(y)&=\\log_a(xy)\\text{,}\\\\
\\log_a(x)-\\log_a(y)&=\\log_a\\left(\\frac{x}{y}\\right)\\text{.}
\\end{align}\\]

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$ \\log_a (\\var{x1[5]}^\\var{x1[6]}) $ = [[0]]$\\log_a$ ([[1]])

When adding and subtracting logarithms we can simplify the expressions using some logarithm laws. These laws are

\\[\\begin{align}
p \\log_a(x) &=\\log_a(x^p)\\text{.}\\\\
\\end{align}\\]

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Questions testing rather basic understanding of the index laws.

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Exponential functions $y = \\log_k (ax).$ Growth or decay, x-y intercepts.

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Identify if the graph $y= \\ln (\\var{m} x)$ is increasing or decreasing.

[[0]]

Does it have an y intercept

[[1]]

What is the x intercept

[[2]]

Domain is (put as $x$, first box for sign (>, >=, < or <=), socond box - value):

$x$[[3]][[4]]

Range is (put in boxes: value (or -infinity), sign (< or <=), $y$, sign (< or <=), value (or infinity):

[[5]][[6]]$y$[[7]][[8]]

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Solve a logarithmic equation

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Solve the following logarithmic equations

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\$\\var{a}log(\\var{b}x+\\var{c})=\\var{d}\$

Divide across by \$\\var{a}\$

\$log(\\var{b}x+\\var{c})=\\var{d}/\\var{a}=\\simplify{{d}/{a}}\$

\$\\var{b}x+\\var{c}=10^{\\simplify{{d}/{a}}}\$

\$\\var{b}x+\\var{c}=\\simplify{10^{{d}/{a}}}\$

\$\\var{b}x=\\simplify{10^{{d}/{a}}}-\\var{c}\$

\$\\var{b}x=\\simplify{10^{{d}/{a}}-{c}}\$

\$x=\\simplify{(10^{{d}/{a}}-{c})/{b}}\$

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Given the following logarithmic equation:

\$y=\\log_\\var{a2} (\\var{b2}x+\\var{c2}))\$

calculate the value of \$x\$ that satisfies the equation when \$y=\\var{d2}\$.

Input your answer correct to three decimal places.

\$x = \$ [[0]]

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