Question 365680
{{{ ln(t/(t-4))=1 }}} Start with the given equation.



{{{ ln(t/(t-4))=ln(e) }}} Rewrite 1 as {{{ln(e)}}}. Recall that {{{ln(e)=1}}}



Since the logs have the same base (e), this means that the arguments (the stuff inside the logs) are equal.



Formally, if {{{ln(x)=ln(y)}}}, then {{{x=y}}}



{{{ t/(t-4)=e }}} Use the property given above to set the arguments equal to each other.



{{{ t=e(t-4) }}} Multiply both sides by {{{t-4}}}.



{{{ t=et-4e }}} Distribute.



{{{ t-et=-4e }}} Subtract et from both sides.



{{{ t(1-e)=-4e }}} Factor out the GCF 't'.



{{{ t=-4e/(1-e) }}} Divide both sides by {{{1-e}}} to isolate 't'.



<font color=red>Optional Step:</font> You can rewrite {{{1-e}}} as {{{-e+1=-(e-1)}}} and notice how the negatives cancel in the numerator and denominator. So {{{t=4e/(e-1)}}} also



So the solution is {{{ t=-4e/(1-e) }}}



You can also write the solution as {{{t=4e/(e-1)}}} (the two are equivalent).



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Jim