Expand the logarithmic expression log((x^5*10^(1/2))/y)

 Exercise

\log\left(\frac{x^5\sqrt{10}}{y}\right)

 

The difference of two logarithms of equal base $b$ is equal to the logarithm of the quotient: $\log_b(x)-\log_b(y)=\log_b\left(\frac{x}{y}\right)$

\log \left(\sqrt{10}x^5\right)-\log \left(y\right)

 

Use the product rule for logarithms: $\log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right)$ , where $M=x^5$ and $N=\sqrt{10}$

\log \left(x^5\right)+\log \left(\sqrt{10}\right)-\log \left(y\right)

 

Use the following rule for logarithms: $\log_b(b^k)=k$

\log \left(x^5\right)+\frac{1}{2}-\log \left(y\right)

 

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

5\log \left(x\right)+\frac{1}{2}-\log \left(y\right)

5\log \left(x\right)+\frac{1}{2}-\log \left(y\right)

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