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Find all solutions of the equation below.

4^(3x+1)=7^(x)
The solution(s) is/are
x=
◻
(Simplify your answer. Use a comma to separate answers as needed. Round to four decimal places as needed.)

Find all solutions of the equation below. $\newline$ $4^{3 \mathrm{x}+1}=7^{x}$ $\newline$ The solution(s) is/are $x=$ $\square$ $\newline$ (Simplify your answer. Use a comma to separate answers as needed. Round to four decimal places as needed.)

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Conjugate root theorems

Full solution

Q. Find all solutions of the equation below.

\newline

4^{3 \mathrm{x}+1}=7^{x}

\newline

The solution(s) is/are

x=

\square

\newline

(Simplify your answer. Use a comma to separate answers as needed. Round to four decimal places as needed.)

Take Logarithm: Take the logarithm of both sides to simplify the equation; using natural logarithm ( $\ln$ ) is common for such equations.
Apply Identity: Apply the logarithmic identity $\ln(a^b) = b\cdot\ln(a)$ to both sides.
Distribute $\ln(4)$ : Distribute $\ln(4)$ on the left side.
Rearrange Equation: Rearrange the equation to isolate terms involving $x$ on one side.
Factor Out $x$ : Factor out $x$ from the left side.
Solve for x: Solve for x by dividing both sides by $(3 \cdot \ln(4) - \ln(7))$ .
Calculate Value: Calculate the value using a calculator for precision, rounding to four decimal places as needed.

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