6.3 Logarithmic functions  (Page 2/9)

First, identify the values of $b,y,\text{and}x.$ Then, write the equation in the form $b^y=x.$

1. ${\log }_6\left(\sqrt{6}\right)=\frac{1}{2}$

   Here, $b=6,y=\frac{1}{2},\text{and} x=\sqrt{6.}$ Therefore, the equation ${\log }_6\left(\sqrt{6}\right)=\frac{1}{2}$ is equivalent to $$6^{\frac{1}{2}}=\sqrt{6}.$$

2. ${\log }_3\left(9\right)=2$

   Here, $b=3,y=2,\text{and} x=9.$ Therefore, the equation ${\log }_3\left(9\right)=2$ is equivalent to $3^2=9.$

First, identify the values of $b,y,\text{and}x.$ Then, write the equation in the form $x={\log }_b\left(y\right).$

1. $2^3=8$

   Here, $b=2,$ $x=3,$ and $y=8.$ Therefore, the equation $2^3=8$ is equivalent to ${\log }_2(8)=3.$

2. $5^2=25$

   Here, $b=5,$ $x=2,$ and $y=25.$ Therefore, the equation $5^2=25$ is equivalent to ${\log }_5(25)=2.$

3. ${10}^{-4}=\frac{1}{\mathrm{10,000}}$

   Here, $b=10,$ $x=-4,$ and $y=\frac{1}{\mathrm{10,000}}.$ Therefore, the equation ${10}^{-4}=\frac{1}{\mathrm{10,000}}$ is equivalent to ${\text{log}}_{10}(\frac{1}{\mathrm{10,000}})=-4.$