Solve the logarithmic equation ln(y)=ln(3x)ln(x^(1/2))

 Exercise

$\ln\left(y\right)=\ln\left(3x\right)\ln\left(\sqrt{x}\right)$

 

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$\ln\left(y\right)=\frac{1}{2}\ln\left(3x\right)\ln\left(x\right)$

 

Multiplying the fraction by $\ln\left(3x\right)\ln\left(x\right)$

$\ln\left(y\right)=\frac{\ln\left(3x\right)\ln\left(x\right)}{2}$

 

Applying the product rule for logarithms: $\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)$

$\ln\left(y\right)=\frac{\left(\ln\left(3\right)+\ln\left(x\right)\right)\ln\left(x\right)}{2}$

 

Take the variable outside of the logarithm

$e^{\ln\left(y\right)}=e^{\frac{\left(\ln\left(3\right)+\ln\left(x\right)\right)\ln\left(x\right)}{2}}$

 

Simplifying the logarithm

$y=e^{\frac{\left(\ln\left(3\right)+\ln\left(x\right)\right)\ln\left(x\right)}{2}}$

$y=e^{\frac{\left(\ln\left(3\right)+\ln\left(x\right)\right)\ln\left(x\right)}{2}}$

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