Solve for x as a power

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Clodagh Carroll

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Clodagh Carroll 1 year, 2 months ago

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Clodagh Carroll 1 year, 2 months ago

Created this as a copy of When the bases don't match (with progressive hints).

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Question statement

Solve for $x$ in the following equation, rounding your answer to $\var{dec}$ decimal places:

$\var{a}^{\simplify{{c}x+{d}}} = \var{b}^{\simplify{{f}x+{g}}}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	7
b	integer	4
c	integer	8
d	integer	1
f	integer	4
g	integer	3
dec	integer	5
x	number	0.2208092223
ans	number	0.22081
addition_or_subtraction_vector	list	[ "ADDITION", "SUBTRACTION" ]
add_or_subtract_vector	list	[ "ADD", "SUBTRACT" ]
n	integer	0
addition_or_subtraction	string	ADDITION
add_or_subtract	string	SUBTRACT
from_or_to_vector	list	[ "to", "from" ]
from_or_to	string	from
d_plus_or_minus_vector	list	[ "+", "-" ]
d_plus_or_minus	string	+
abs_d	number	1
g_plus_or_minus_vector	list	[ "+", "-" ]
m	integer	0
g_plus_or_minus	string	+
abs_g	number	3
not_d_plus_or_minus	string	-

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Base of number on left-hand side of equation.

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Beginning with the equation we are given $\var{a}^{\simplify{{c}x+{d}}} = \var{b}^{\simplify{{f}x+{g}}}$ ,

note the following rule of logs:

$\hspace{0.5cm} \begin{align*} \log_a (x^p) &= p \cdot \log _a (x)\\ \\ \text{e.g., } \log_3 (7^4) & = 4 \cdot \log_3(7) \end{align*} \hspace{0.5cm}$

Therefore, if we introduce logs, we can use this rule to get rid of the power of $\simplify{{c}x+{d}}$ on the left and the power of $\simplify{{f}x+{g}}$ on the right!

Since all calculators have a $\log_{10}$ button (usually just written as $\log$ on the calculator), we will use $\log_{10}$ to do this:

$\begin{align*} \var{a}^{\simplify{{c}x+{d}}} & = \var{b}^{\simplify{{f}x+{g}}}\\ \log_{10} ( \var{a}^{\simplify{{c}x+{d}}} ) & = \log_{10} ( \var{b}^{\simplify{{f}x+{g}}} ) \hspace{2cm} \{\textbf{Take } \log_{10} \textbf{ of both sides.} \} \end{align*}$

Now use the rule of logs mentioned in the box above to bring the powers out from the log terms:

$\begin{align*} \log_{10} \left( \var{a}^{\simplify{{c}x+{d}}} \right) & = \log_{10} \left(\var{b}^{\simplify{{f}x+{g}}} \right)\\ \\ (\simplify{{c}x+{d}}) \log_{10} \var{a} & = (\simplify{{f}x+{g}}) \log_{10} \var{b} \end{align*}$

Next, $\textbf{MULTIPLY OUT}$ the brackets on both sides:

$\begin{align*} (\simplify{{c}x+{d}}) \log_{10} \var{a} & = (\simplify{{f}x+{g}}) \log_{10} \var{b}\\ \\ \simplify[!unitFactor]{{c}x} \log_{10} \var{a} \ \ \var{d_plus_or_minus} \ \ \simplify{{abs_d}} \log_{10} \var{a} & = \simplify{{f}x} \log_{10} \var{b}\ \ \var{g_plus_or_minus} \ \ \var{abs_g} \log_{10} \var{b} \hspace{2cm}\{ \textbf{Multiply out the brackets}\}\end{align*}$

PLAN:

Group all $x$ -terms on one side

and all number terms on the other side.

BE VERY CAREFUL WITH THE SIGNS HERE!

So let's make sure all terms that have an $x$ are on the left-hand side of the equation and all terms without an $x$ are on the right-hand side.

To cancel the $\textbf{ADDITION}$ of $\simplify{{f}x} \log_{10} \var{b}$ on the right-hand side, we must $\textbf{SUBTRACT }$ $\simplify{{f}x} \log_{10} \var{b}$ from both sides:

$\begin{align*} \simplify[!unitFactor]{{c}x} \log_{10} \var{a} \ \ \var{d_plus_or_minus} \ \ \simplify{{abs_d}} \log_{10} \var{a} & = \simplify{{f}x} \log_{10} \var{b}\ \ \var{g_plus_or_minus} \ \ \var{abs_g} \log_{10} \var{b} \\ \\ \hspace{-6cm} -\simplify{{f}x} \log_{10} \var{b} \bigg\downarrow \phantom{\hspace{2.5cm}} & \hspace{0.75cm} \bigg\downarrow -\simplify{{f}x} \log_{10} \var{b}\\ \\ \simplify[!unitFactor]{{c}x} \log_{10} \var{a} \ \ \var{d_plus_or_minus} \ \ \simplify{{abs_d}} \log_{10} \var{a} - \simplify{{f}x} \log_{10} \var{b} & = \var{g} \log_{10} \var{b} \end{align*}$

To cancel the $\var{addition_or_subtraction}$ of $\simplify{(-1)^({n}) {d}} \log_{10} \var{a}$ on the left-hand side, we must $\var{add_or_subtract}$ $\simplify{(-1)^({n}) {d}} \log_{10} \var{a}$ $\var{from_or_to}$ both sides:

$\begin{align*} \simplify[!unitFactor]{{c}x} \log_{10} \var{a} \ \ \var{d_plus_or_minus} \ \ \simplify{{abs_d}} \log_{10} \var{a} - \simplify{{f}x} \log_{10} \var{b} & = \var{g} \log_{10} \var{b} \\ \\ \hspace{-6cm} \var{not_d_plus_or_minus} \ \simplify{(-1)^({n}) {d}} \log_{10} \var{a} \bigg\downarrow \phantom{\hspace{2.5cm}} & \hspace{0.75cm} \bigg\downarrow \var{not_d_plus_or_minus} \ \simplify{(-1)^({n}) {d}} \log_{10} \var{a}\\ \\ \simplify[!unitFactor]{{c}x} \log_{10} \var{a} - \simplify{{f}x} \log_{10} \var{b} & = \var{g} \log_{10} \var{b} \ \ \var{not_d_plus_or_minus} \ \ \simplify{(-1)^({n}) {d}} \log_{10} \var{a} \end{align*}$

Now, note that both terms on the left-hand side have an $x$ in common:

$\begin{align*} [\var{c} \log_{10} \var{a}] x - [\var{f} \log_{10} \var{b}] x & = \var{g} \log_{10} \var{b} \ \ \var{not_d_plus_or_minus} \ \ \simplify{(-1)^({n}) {d}} \log_{10} \var{a}\\ \\ \bigg\downarrow \hspace{0.75cm} \textbf{Factor out } & \textbf{the common } x \textbf{ term} \\ \\ [\var{c} \log_{10} \var{a} - \var{f} \log_{10} \var{b}] x & = \var{g} \log_{10} \var{b} \ \ \var{not_d_plus_or_minus} \ \ \simplify{(-1)^({n}) {d}} \log_{10} \var{a} \end{align*}$

To cancel the $\textbf{MULTIPLICATION}$ of $\var{c} \log_{10} \var{a} - \var{f} \log_{10} \var{b}$ on the left-hand side, we must $\textbf{DIVIDE}$ both sides by $\var{c} \log_{10} \var{a} - \var{f} \log_{10} \var{b}$ :

$\begin{align*} [\var{c} \log_{10} \var{a} - \var{f} \log_{10} \var{b}] x & = \var{g} \log_{10} \var{b} \ \ \var{not_d_plus_or_minus} \ \ \simplify{(-1)^({n}) {d}} \log_{10} \var{a} \\ \\ \div \ \big[ \var{c} \log_{10} \var{a} - \var{f} \log_{10} \var{b} \big] \bigg\downarrow \phantom{\hspace{2.5cm}} & \hspace{0.75cm} \bigg\downarrow \div \ \big[ \var{c} \log_{10} \var{a} - \var{f} \log_{10} \var{b} \big]\\ \\ x & = \frac{\var{g} \log_{10} \var{b} \ \ \var{not_d_plus_or_minus} \ \ \simplify{(-1)^({n}) {d}} \log_{10} \var{a}}{\var{c} \log_{10} \var{a} - \var{f} \log_{10} \var{b}} \\ \\ & \hspace{0.75cm} \bigg\downarrow \{ \textbf{Calculator work} \}\\ \\ x & =\var{x}\\ \\ & \hspace{0.75cm} \bigg\downarrow \textbf{Round to } \var{dec} \textbf{ decimal places} \\ \\ x & = \var{ans} \end{align*}$

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Clodagh Carroll 1 year, 2 months ago

Clodagh Carroll 1 year, 2 months ago

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Solve for x as a power

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

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Checkpoint description

Clodagh Carroll 1 year, 2 months ago

Clodagh Carroll 1 year, 2 months ago

Error in variable testing condition

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Generated value: integer

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Objectives

Penalties

Generated value: `integer`