Solve the following for x 2^(2x+1)=7^(x)

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Solve the following for  x  2^(2x+1)=7^(x)

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Write Equation: To solve the equation 2^(2x+1) = 7^(x), we need to find a way to compare the exponents of the same base or use logarithms to solve for x. First, let's write down the equation: 2^(2x+1) = 7^(x)Apply Logarithms: Since the bases are different and there is no simple way to write them with a common base, we will use logarithms to solve for x. We can take the natural logarithm (ln) of both sides of the equation to utilize the property that ln(a^b) = b*ln(a). Taking the natural logarithm of both sides gives us: ln(2^(2x+1)) = ln(7^(x))Use Logarithm Property: Using the property of logarithms that ln(a^b) = b*ln(a), we can bring down the exponents in front of the logarithms: (2x+1)*ln(2) = x*ln(7)Linear Equation in x: Now we have a linear equation in terms of x. We can distribute ln(2) on the left side to separate the terms involving x: 2x*ln(2) + ln(2) = x*ln(7)Factor Out x: Next, we want to get all the terms involving x on one side of the equation and constants on the other side. We can subtract x*ln(7) from both sides to achieve this: 2x*ln(2) - x*ln(7) = -ln(2)Divide to Solve: Now we can factor out x from the left side of the equation: x*(2*ln(2) - ln(7)) = -ln(2)Calculate x: To solve for x, we divide both sides of the equation by (2*ln(2) - ln(7)): x = -ln(2) / (2*ln(2) - ln(7))Finally, we can calculate the value of x using a calculator: x ≈ -ln(2) / (2*ln(2) - ln(7))

Solve the following for `x` $\newline$ $2^{(2x+1)}=7^{x}$

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Solve the following for `x`\newline2(2x+1)=7x2^{(2x+1)}=7^{x}2(2x+1)=7x

Solve the following for `x` $\newline$ $2^{(2x+1)}=7^{x}$