Answer:
[tex]\log 8 - \log x + 7\log\sqrt x =\log (8x^{\frac{5}{2}})[/tex]
Step-by-step explanation:
Given
[tex]\log 8 - \log x + 7\log\sqrt x[/tex]
Required
Express as a single expression
We have:
[tex]\log 8 - \log x + 7\log\sqrt x[/tex]
Write 7 as an exponent
[tex]\log 8 - \log x + 7\log\sqrt x =\log 8 - \log x + \log(\sqrt x)^7[/tex]
Rewrite as:
[tex]\log 8 - \log x + 7\log\sqrt x =\log 8 - \log x + \log(x^{\frac{1}{2}})^7[/tex]
[tex]\log 8 - \log x + 7\log\sqrt x =\log 8 - \log x + \log x^\frac{7}{2}[/tex]
Apply quotient and product rule of logarithm
[tex]\log 8 - \log x + 7\log\sqrt x =\log (\frac{8*x^\frac{7}{2}}{x} )[/tex]
Apply law of indices
[tex]\log 8 - \log x + 7\log\sqrt x =\log (8*x^{\frac{7}{2} - 1})[/tex]
Solve exponent
[tex]\log 8 - \log x + 7\log\sqrt x =\log (8*x^{\frac{7-2}{2}})[/tex]
[tex]\log 8 - \log x + 7\log\sqrt x =\log (8*x^{\frac{5}{2}})[/tex]
[tex]\log 8 - \log x + 7\log\sqrt x =\log (8x^{\frac{5}{2}})[/tex]
