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

2 Answers
Feb 22, 2018

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)

Feb 22, 2018

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)