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Formula
Solve the equation
Answer
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$3 ^{ 5 } +3 ^{ 5 } +3 ^{ 5 } = 3 ^{ a }$
$a = 6$
Solve the equation
$\color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } = \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ a } }$
$ $ Invert the left and right terms to solve the exponential equation (inequality) $ $
$\color{#FF6800}{ a } = \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \right) }$
$a = \log _{ 3 } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \color{#FF6800}{ + } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \right) }$
$ $ Add the forms of the powers with the same bases and exponents $ $
$a = \log _{ 3 } { \left( \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \right) }$
$a = \log _{ 3 } { \left( \color{#FF6800}{ 3 } \color{#FF6800}{ \times } \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 5 } } \right) }$
$ $ Simplify the expression $ $
$a = \log _{ 3 } { \left( \color{#FF6800}{ 729 } \right) }$
$a = \log _{ 3 } { \left( \color{#FF6800}{ 729 } \right) }$
$ $ Write the number in exponential form with base $ 3$
$a = \log _{ 3 } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 6 } } \right) }$
$a = \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } ^ { \color{#FF6800}{ 6 } } \right) }$
$ $ Simplify the expression using $ \log_{a}{a^{x}}=x\times\log_{a}{a}$
$a = \color{#FF6800}{ 6 } \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } \right) }$
$a = 6 \log _{ \color{#FF6800}{ 3 } } { \left( \color{#FF6800}{ 3 } \right) }$
$ $ The logarithm is equal to 1 if a base is same as an antilogarithm $ $
$a = 6 \times \color{#FF6800}{ 1 }$
$a = 6 \color{#FF6800}{ \times } \color{#FF6800}{ 1 }$
$ $ Multiplying any number by 1 does not change the value $ $
$a = \color{#FF6800}{ 6 }$
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