Given: The logarithm below
[tex]log_7\sqrt{3}[/tex]
To Determine: The solution of the given
Solution
Using exponent rule below
[tex]\sqrt{a}=a^{\frac{1}{2}}[/tex]
Applying the exponent rule above to the given logarithm
[tex]\begin{gathered} log_7\sqrt{3} \\ \sqrt{3}=3^{\frac{1}{2}} \\ Therefore \\ log_7\sqrt{3}=log_73^{\frac{1}{2}} \end{gathered}[/tex]
Using logarithm rule to the given
[tex]\begin{gathered} log_ab^x=xlog_ab \\ Therefore \\ log_73^{\frac{1}{2}}=\frac{1}{2}log_73 \end{gathered}[/tex]
Let us apply change of base as shown below
[tex]log_73=\frac{log_e3}{log_e7}[/tex][tex]\begin{gathered} Log_e3=ln3=1.09861 \\ log_e7=ln7=1.94591 \\ log_73=\frac{log_e3}{log_e7}=\frac{ln3}{ln7}=\frac{1.09861}{1.94591}=0.56457 \end{gathered}[/tex]
Therefore, we have
[tex]log_7\sqrt{3}=\frac{1}{2}log_73=\frac{1}{2}\times0.56457=0.28228\approx0.2823(4decimal-place)[/tex]
Hence, the final answer is approximately 0.2823
