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Formula
Solve the equation
Answer
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Graph
$y = 3 ^ { 2 } \times 81$
$y = 3 ^ { x + 1 }$
$y$Intercept
$\left ( 0 , 3 \right )$
Asymptote
$y = 0$
$3 ^{ 2 } \times 81 = 3 ^{ x+1 }$
$x = 5$
Solve the equation
$\color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 81 } = \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } }$
$ $ Invert the left and right terms to solve the exponential equation (inequality) $ $
$\color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 81 } \right) }$
$x + 1 = \log _{ 3 } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ \times } \color{#FF6800}{ 81 } \right) }$
$ $ Simplify the expression $ $
$x + 1 = \log _{ 3 } { \left( \color{#FF6800}{ 9 } \color{#FF6800}{ \times } \color{#FF6800}{ 81 } \right) }$
$x + 1 = \log _{ 3 } { \left( \color{#FF6800}{ 9 } \color{#FF6800}{ \times } \color{#FF6800}{ 81 } \right) }$
$ $ Multiply $ 9 $ and $ 81$
$x + 1 = \log _{ 3 } { \left( \color{#FF6800}{ 729 } \right) }$
$x + 1 = \log _{ 3 } { \left( \color{#FF6800}{ 729 } \right) }$
$ $ Write the number in exponential form with base $ 3$
$x + 1 = \log _{ 3 } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 6 } } \right) }$
$x + 1 = \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 6 } } \right) }$
$ $ Simplify the expression using $ \log_{a}{a^{x}}=x\times\log_{a}{a}$
$x + 1 = \color{#FF6800}{ 6 } \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } \right) }$
$x + 1 = 6 \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } \right) }$
$ $ The logarithm is equal to 1 if a base is same as an antilogarithm $ $
$x + 1 = 6 \times \color{#FF6800}{ 1 }$
$x + 1 = 6 \color{#FF6800}{ \times } \color{#FF6800}{ 1 }$
$ $ Multiplying any number by 1 does not change the value $ $
$x + 1 = \color{#FF6800}{ 6 }$
$x \color{#FF6800}{ + } \color{#FF6800}{ 1 } = 6$
$ $ Move the constant to the right side and change the sign $ $
$x = 6 \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
$x = \color{#FF6800}{ 6 } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
$ $ Subtract $ 1 $ from $ 6$
$x = \color{#FF6800}{ 5 }$
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