Question

Rewrite the following in the form log(c)? Log(2)+log(4)

Answer 1

`log(8)`

Explanation:

We can use the product property for logarithms:

`log(a)+log(b)=log(ab)`

In our case, `a=2` and `b=4`. We can multiply `a` and `b` now to get the new number we're taking the logarithm of.

`log(2)+log(4)=log(2*4)=log(8)`

Answer 2

`log(2) + log(4) = log(8)`

Explanation:

We may first note that `log(2) + log(4)` actually looks just like something we would get after expanding the logarithm of a product! You may be familiar with the following rule:

`log_b(uv) = log_b(u) + log_b(v)`

We derive this by starting with a product.

`color(red)(uv) = color(green)ucolor(blue)v`

By definition of the logarithm function, we can raise `b` to the power of `log_b` of some value, and still have the original value.

`b^(log_b(color(red)(uv))) = b^(log_b(color(green)u)) * b^(log_b(color(blue)u))`

`b^(log_b(uv)) = b^(log_b(u) + log_b(v))`

And since these have the same base, we can conclude that the exponents must be equal.

`log_b(uv) = log_b(u) + log_b(v)`

Now, for this problem in particular, we are working backward.

`log(c) = log(2) + log(4)`

Doing a bit of pattern matching, we notice that `c` must be the product of `2` and `4`.

`log(2*4) = log(2) + log(4)`

`log(8) = log(2) + log(4)`