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Second Fundamental Theorem of Calculus

If 2x.3x+4 ...

Question

If $2^{x} {.3}^{x + 4} = 7^{x},$ then $x =$

\frac{4 l o g_{e} 3}{l o g_{e} 7 - l o g_{e} 6}

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\frac{4 l o g_{e} 3}{l o g_{e} 6 - l o g_{e} 7}

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\frac{2 l o g_{e} 3}{l o g_{e} 7 - l o g_{e} 6}

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\frac{3 l o g_{e} 3}{l o g_{e} 6 - l o g_{e} 7}

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Solution

The correct option is A $\frac{4 l o g_{e} 3}{l o g_{e} 7 - l o g_{e} 6}$
$2^{x} {.3}^{x + 4} = 7^{x} ⟹ {log}_{e} 2^{x} + {log}_{e} 3^{x + 4} = {log}_{e} 7^{x}$
$x {log}_{e} 2 + (x + 4) {log}_{e} 3 = x {log}_{e} 7$
$4 {log}_{e} 3 = x ({log}_{e} 7 - {log}_{e} 3 - {log}_{e} 2)$
$x = \frac{4 {log}_{e} 3}{{log}_{e} 7 - {log}_{e} 6}$

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