Question

(Solve the problem & round the final answer to four decimal places.)

Answer 1

Given: The logarithm below

$log_7\sqrt{3}$

To Determine: The solution of the given

Solution

Using exponent rule below

$\sqrt{a}=a^{\frac{1}{2}}$

Applying the exponent rule above to the given logarithm

$\begin{gathered} log_7\sqrt{3} \\ \sqrt{3}=3^{\frac{1}{2}} \\ Therefore \\ log_7\sqrt{3}=log_73^{\frac{1}{2}} \end{gathered}$

Using logarithm rule to the given

$\begin{gathered} log_ab^x=xlog_ab \\ Therefore \\ log_73^{\frac{1}{2}}=\frac{1}{2}log_73 \end{gathered}$

Let us apply change of base as shown below

$log_73=\frac{log_e3}{log_e7}$$\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}$

Therefore, we have

$log_7\sqrt{3}=\frac{1}{2}log_73=\frac{1}{2}\times0.56457=0.28228\approx0.2823(4decimal-place)$

Hence, the final answer is approximately 0.2823