csat-suneung 2022 Q13

csat-suneung · South-Korea · csat__math 4 marks Laws of Logarithms Determine Parameters of a Logarithmic Function

For two constants $a , b$ with $1 < a < b$, the $y$-intercept of the line passing through the two points $\left( a , \log _ { 2 } a \right) , \left( b , \log _ { 2 } b \right)$ and the $y$-intercept of the line passing through the two points $\left( a , \log _ { 4 } a \right) , \left( b , \log _ { 4 } b \right)$ are equal.
For the function $f ( x ) = a ^ { b x } + b ^ { a x }$ with $f ( 1 ) = 40$, what is the value of $f ( 2 )$? [4 points]
(1) 760
(2) 800
(3) 840
(4) 880
(5) 920

For two constants $a , b$ with $1 < a < b$, the $y$-intercept of the line passing through the two points $\left( a , \log _ { 2 } a \right) , \left( b , \log _ { 2 } b \right)$ and the $y$-intercept of the line passing through the two points $\left( a , \log _ { 4 } a \right) , \left( b , \log _ { 4 } b \right)$ are equal.

For the function $f ( x ) = a ^ { b x } + b ^ { a x }$ with $f ( 1 ) = 40$, what is the value of $f ( 2 )$? [4 points]\\
(1) 760\\
(2) 800\\
(3) 840\\
(4) 880\\
(5) 920

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