Question

I'm having some problems with this logarithmic question I will upload a photo

Answer 1

The Solution.

`Let\log _{(1)/(9)}((1)/(9))=x`

Writing the above equation in index form, we have

`\begin{gathered} ((1)/(9))^1=((1)/(9))^x \\ Then\text{ it follows that} \\ 1=x \\ x=1 \end{gathered}`
`\begin{gathered} \text{Let }\log _749=x \\ So\text{ we have} \\ 49=7^x \\ \text{Making the base of both sides equal, we have} \\ 7^2=7^x \\ x=2 \end{gathered}`
`\begin{gathered} \text{Let }\log _{(1)/(4)}16=x \\ \\ 16=((1)/(4))^x \\ 16=4^(-1* x) \\ 4^2=4^(-x) \\ -x=2 \\ x=-2 \end{gathered}`
`\begin{gathered} \text{Let }\log _(125)5=x \\ \text{cross multiplying, we have} \\ 5=125^x \\ 5^1=5^(3x) \\ 3x=1 \\ \text{Dividing both sides by 3, we get} \\ x=(1)/(3) \end{gathered}`
`\begin{gathered} \text{ Let }\log _8((1)/(8))=x \\ \\ (1)/(8)=8^x \\ \\ 8^(-1)=8^x \\ -1=x \\ x=-1 \end{gathered}`
`\begin{gathered} \text{ Let }\log _9(1)=x \\ 1=9^x \\ 9^0=9^x \\ x=0 \end{gathered}`
`\begin{gathered} \text{ Let }\log _{(1)/(9)}(-1)=x \\ \\ -1=((1)/(9))^x \\ \\ No\text{ solution because it has no real value.} \end{gathered}`