Question

Question #e10e6

Answer 1

`m=-100`

Explanation:

The only way to get the answer

`m = -100`

is to have

`log(-m) + 2 = 4`

as your starting equation. As you know, the common log, which is denoted `log`, is actually the log base `10`. This means that your starting equation can be rewritten as

`log_10 (-m) + 2 = 4`

The first thing to do here is isolate the log on one side of the equation by adding `-2` to both sides

`log_10 (-m) + color(red)(cancel(color(black)(2))) - color(red)(cancel(color(black)(2)))= 4 -2`

`log_10(-m) = 2`

The log function is actually undefined for negative numbers when working with real numbers. This tells you that `-m >= 0`, which implies that `m <= 0`.

Now, the log function is the inverse operation to exponentiation. This means that you're looking for a number that is equal to the base, which in your case is `color(red)(10)`, raised to the power of the result, which is `color(blue)(2)`.

`log_color(red)(10)(color(blue)(-m)) = color(darkgreen)(2)`

Can thus be rewritten as

`color(red)(10)^color(darkgreen)(2) = color(blue)(-m)`

Since `10^2 = 100`, you will have

`100 = -m implies m = color(green)(|bar(ul(color(white)(a/a)color(black)(-100)color(white)(a/a)|)))`

As predicted, `m<=0`.