Question

Find the value of $x:2^x\times 3^{x+4}=7^x$.

Answer 1

Taking the logarithm of both sides of the function.

$$\begin{gathered} \begin{array}{rcl}& \Rightarrow & \log \left(2^x\right)+\log \left(3^{x+4}\right)=\log \left(7^x\right)\\ & \Rightarrow & x\log \left(2\right)+\left(x+4\right)\log \left(3\right)=x\log \left(7\right)\because \left[\log \left(a^m\right)=m\log \left(a\right)\right]\\ & \Rightarrow & x\log \left(2\right)+x\log \left(3\right)+4\log \left(3\right)=x\log \left(7\right)\\ & \Rightarrow & x\left[\log \left(7\right)-\log \left(2\right)-\log \left(3\right)\right]=4\log \left(3\right)\\ & \Rightarrow & x=\frac{4\log \left(3\right)}{\log \left(7\right)-\log \left(2\right)-\log \left(3\right)}\\ & \Rightarrow & x=\frac{4\times 0.4771}{0.8450-0.3010-0.4771}\end{array} \\ \Rightarrow x=28.507 \end{gathered}$$

Hence, the required final answer is $28.507$.