Solve for q a^(q-r)*b^(r-p)*c^(p-q)=1

Math
aq-r⋅br-p⋅cp-q=1
Divide each term by br-p and simplify.
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Divide each term in aq-r⋅br-p⋅cp-q=1 by br-p.
aq-r⋅br-p⋅cp-qbr-p=1br-p
Cancel the common factor of br-p.
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Cancel the common factor.
aq-r⋅br-p⋅cp-qbr-p=1br-p
Divide aq-r⋅cp-q by 1.
aq-r⋅cp-q=1br-p
aq-rcp-q=1br-p
aq-rcp-q=1br-p
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
ln(aq-rcp-q)=ln(1br-p)
Expand the left side.
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Rewrite ln(aq-rcp-q) as ln(aq-r)+ln(cp-q).
ln(aq-r)+ln(cp-q)=ln(1br-p)
Expand ln(aq-r) by moving q-r outside the logarithm.
(q-r)ln(a)+ln(cp-q)=ln(1br-p)
Expand ln(cp-q) by moving p-q outside the logarithm.
(q-r)ln(a)+(p-q)ln(c)=ln(1br-p)
(q-r)ln(a)+(p-q)ln(c)=ln(1br-p)
Expand the right side.
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Rewrite ln(1br-p) as ln(1)-ln(br-p).
(q-r)ln(a)+(p-q)ln(c)=ln(1)-ln(br-p)
Expand ln(br-p) by moving r-p outside the logarithm.
(q-r)ln(a)+(p-q)ln(c)=ln(1)-((r-p)ln(b))
The natural logarithm of 1 is 0.
(q-r)ln(a)+(p-q)ln(c)=0-((r-p)ln(b))
Subtract (r-p)ln(b) from 0.
(q-r)ln(a)+(p-q)ln(c)=-((r-p)ln(b))
(q-r)ln(a)+(p-q)ln(c)=-((r-p)ln(b))
Simplify each term.
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Apply the distributive property.
qln(a)-rln(a)+(p-q)ln(c)=-((r-p)ln(b))
Apply the distributive property.
qln(a)-rln(a)+pln(c)-qln(c)=-((r-p)ln(b))
qln(a)-rln(a)+pln(c)-qln(c)=-((r-p)ln(b))
Simplify -((r-p)ln(b)).
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Apply the distributive property.
qln(a)-rln(a)+pln(c)-qln(c)=-(rln(b)-pln(b))
Apply the distributive property.
qln(a)-rln(a)+pln(c)-qln(c)=-(rln(b))-(-pln(b))
Multiply -(-pln(b)).
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Multiply -1 by -1.
qln(a)-rln(a)+pln(c)-qln(c)=-rln(b)+1(pln(b))
Multiply p by 1.
qln(a)-rln(a)+pln(c)-qln(c)=-rln(b)+pln(b)
qln(a)-rln(a)+pln(c)-qln(c)=-rln(b)+pln(b)
qln(a)-rln(a)+pln(c)-qln(c)=-rln(b)+pln(b)
Move all the terms containing a logarithm to the left side of the equation.
qln(a)-rln(a)+pln(c)-qln(c)+rln(b)-pln(b)=0
Move all terms not containing q to the right side of the equation.
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Add rln(a) to both sides of the equation.
qln(a)+pln(c)-qln(c)+rln(b)-pln(b)=rln(a)
Subtract pln(c) from both sides of the equation.
qln(a)-qln(c)+rln(b)-pln(b)=rln(a)-pln(c)
Subtract rln(b) from both sides of the equation.
qln(a)-qln(c)-pln(b)=rln(a)-pln(c)-rln(b)
Add pln(b) to both sides of the equation.
qln(a)-qln(c)=rln(a)-pln(c)-rln(b)+pln(b)
qln(a)-qln(c)=rln(a)-pln(c)-rln(b)+pln(b)
Factor q out of qln(a)-qln(c).
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Factor q out of qln(a).
q(ln(a))-qln(c)=rln(a)-pln(c)-rln(b)+pln(b)
Factor q out of -qln(c).
q(ln(a))+q(-1ln(c))=rln(a)-pln(c)-rln(b)+pln(b)
Factor q out of qln(a)+q(-1ln(c)).
q(ln(a)-1ln(c))=rln(a)-pln(c)-rln(b)+pln(b)
q(ln(a)-1ln(c))=rln(a)-pln(c)-rln(b)+pln(b)
Rewrite -1ln(c) as -ln(c).
q(ln(a)-ln(c))=rln(a)-pln(c)-rln(b)+pln(b)
Divide each term by ln(a)-ln(c) and simplify.
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Divide each term in q(ln(a)-ln(c))=rln(a)-pln(c)-rln(b)+pln(b) by ln(a)-ln(c).
q(ln(a)-ln(c))ln(a)-ln(c)=rln(a)ln(a)-ln(c)+-pln(c)ln(a)-ln(c)+-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c)
Cancel the common factor of ln(a)-ln(c).
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Cancel the common factor.
q(ln(a)-ln(c))ln(a)-ln(c)=rln(a)ln(a)-ln(c)+-pln(c)ln(a)-ln(c)+-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c)
Divide q by 1.
q=rln(a)ln(a)-ln(c)+-pln(c)ln(a)-ln(c)+-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c)
q=rln(a)ln(a)-ln(c)+-pln(c)ln(a)-ln(c)+-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c)
Simplify rln(a)ln(a)-ln(c)+-pln(c)ln(a)-ln(c)+-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c).
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Simplify each term.
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Move the negative in front of the fraction.
q=rln(a)ln(a)-ln(c)-pln(c)ln(a)-ln(c)+-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c)
Move the negative in front of the fraction.
q=rln(a)ln(a)-ln(c)-pln(c)ln(a)-ln(c)-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c)
q=rln(a)ln(a)-ln(c)-pln(c)ln(a)-ln(c)-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c)
Combine into one fraction.
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Combine the numerators over the common denominator.
q=rln(a)-pln(c)ln(a)-ln(c)-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c)
Combine the numerators over the common denominator.
q=rln(a)-pln(c)-rln(b)ln(a)-ln(c)+pln(b)ln(a)-ln(c)
Combine the numerators over the common denominator.
q=rln(a)-pln(c)-rln(b)+pln(b)ln(a)-ln(c)
q=rln(a)-pln(c)-rln(b)+pln(b)ln(a)-ln(c)
q=rln(a)-pln(c)-rln(b)+pln(b)ln(a)-ln(c)
q=rln(a)-pln(c)-rln(b)+pln(b)ln(a)-ln(c)
Solve for q a^(q-r)*b^(r-p)*c^(p-q)=1

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