Question

Hi, need help with solving this logarithm.​

Answer 1

`\log 8 - \log x + 7\log\sqrt x =\log (8x^{(5)/(2)})`

Explanation:

Given

`\log 8 - \log x + 7\log\sqrt x`

Required

Express as a single expression

We have:

`\log 8 - \log x + 7\log\sqrt x`

Write 7 as an exponent

`\log 8 - \log x + 7\log\sqrt x =\log 8 - \log x + \log(\sqrt x)^7`

Rewrite as:

`\log 8 - \log x + 7\log\sqrt x =\log 8 - \log x + \log(x^{(1)/(2)})^7`

`\log 8 - \log x + 7\log\sqrt x =\log 8 - \log x + \log x^(7)/(2)`

Apply quotient and product rule of logarithm

`\log 8 - \log x + 7\log\sqrt x =\log ((8*x^(7)/(2))/(x) )`

Apply law of indices

`\log 8 - \log x + 7\log\sqrt x =\log (8*x^{(7)/(2) - 1})`

Solve exponent

`\log 8 - \log x + 7\log\sqrt x =\log (8*x^{(7-2)/(2)})`

`\log 8 - \log x + 7\log\sqrt x =\log (8*x^{(5)/(2)})`

`\log 8 - \log x + 7\log\sqrt x =\log (8x^{(5)/(2)})`