Question 955429
(1) {{{ x = log( a^2*b^3 ) }}}
(2) {{{ y = log( a/b ) }}}
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(1) {{{ x = log( a^2 ) + log( b^3 ) }}}
(1) {{{ x = 2*log(a) + 3*log(b) }}}
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(2) {{{ y = log(a) - log(b) }}}
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Multiply both sides of (2) by {{{ 3 }}}
and add the equations
(1) {{{ x = 2*log(a) + 3*log(b) }}}
(2) {{{ 3y = 3*log(a) - 3*log(b) }}}
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{{{ x + 3y = 5*log(a) }}}
{{{ log(a) = ( x + 3y ) / 5 }}}
and
(2) {{{ y = log(a) - log(b) }}}
(2) {{{ y = ( x + 3y ) / 5 - log(b) }}}
(2) {{{ log(b) = ( x + 3y ) / 5 - y }}}
(2) {{{ log(b) = x/5 + (3y)/5 - (5*y)/5 }}}
(2) {{{ log(b) = x/5 - (2y)/5 }}}
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{{{ log(a) = ( x + 3y ) / 5 }}}
{{{ log(b) = ( x - 2y ) / 5 }}} answers
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check:
Let {{{ a = 10 }}}
Let {{{ b = 100 }}}
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(1) {{{ x = log( a^2*b^3 ) }}}
(1) {{{ x = log( 10^2*100^3 ) }}}
(1) {{{ x = log( 10^8 ) }}}
(1) {{{ x = 8 }}}
and
(2) {{{ y = log( a/b ) }}}
(2) {{{ y = log( 10/100 ) }}}
(2) {{{ y = log( 10^(-1) ) }}}
(2) {{{ y = -1 }}}
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{{{ log(a) = ( x + 3y ) / 5 }}}
{{{ log(10) = ( 8 - 3 ) / 5 }}}
{{{ log(10) = 5/5 }}}
{{{ 1 = 1 }}}
OK
{{{ log(b) = ( x - 2y ) / 5 }}}
{{{ log(100) = ( 8 - 2*(-1) ) / 5 }}}
{{{ 2 = ( 8+2 ) / 5 }}}
{{{ 2 = 10/5 }}}
{{{ 2 = 2 }}}
OK