Question

Condense into a single logarithm and simplify:

`2\log _(10)x+\log _(10)5`

Answer 1

`2\, \log_(10)(x) + \log_(10)(5) = \log_(10)(5\, x^(2))`.

Explanation:

For any given base
`b > 0`,
`x > 0`, and
`y > 0`:

`\log_(b)(x) + \log_(b)(y) = \log_(b)(x\, y)`.

For all real
`y` (including both
`y > 0` and
`y \le 0`,) as long as
`b > 0`,
`x > 0`:

`y\, \log_(b)(x) = \log_(b)(x^(y))`.

Apply the second rule to rewrite
`2\, \log_(10)(x)`:

`\begin{aligned}2\, \log_(10)(x) &= \log_(10)(x^(2))\end{aligned}`.

Apply the first rule:

`\begin{aligned}\log_(10)(x^(2)) + \log_(10)(5) = \log_(10)(5\, x^(2))\end{aligned}`.

Overall:

`\begin{aligned}& 2\, \log_(10)(x) + \log_(10)(5)\\ =\; & \log_(10)(x^(2)) + \log_(10)(5) \\ =\; & \log_(10)(5\, x^(2))\end{aligned}`.