Question

Which is 3 logx + 4 log(x-2) written as a single logarithm?

logx (x - 2)
logx(x-2)
b. 12 logx(x - 2)
d. 12 logx(x-2)

Answer 1

The answer is
`12\log{(x(x-2))}`

Explanation:

Exponential property of logarithm:

We have that:

`a \log{x} = \log{x^(a)}`

Sum of logarithms:

We have that:

`\log{a} + \log{b} = \log{ab}`

Applying the exponential property:

`3\log{x} = \log{x^3}`

`4\log{(x-2)} = \log{(x-2)^4}`

So

`3\log{x} + 4\log{x-2} = \log{x^3} + \log{(x-2)^4}`

Additive property

`\log{x^3} + \log{(x-2)^4} = \log{x^3(x-2)^4} = \log{(x(x-2))^12}`

Exponential property:

`\log{(x(x-2))^12} = 12\log{(x(x-2))}`

The answer is
`12\log{(x(x-2))}`