Domain with natural logs

**Maskawisewin** · June 24th 2012, 02:40 PM

I'm working with: $f(x)=ln(x)-ln(1-4x)$ and I need to figure out the domain of this function. I know that natural logs cannot be zero or negative numbers but is there a trick to simplying this to seeing the domain easily? Thanks.

**Reckoner** · June 24th 2012, 02:47 PM

Originally Posted by Maskawisewin

I'm working with: $f(x)=ln(x)-ln(1-4x)$ and I need to figure out the domain of this function. I know that natural logs cannot be zero or negative numbers...

You said yourself that we cannot take the log of a nonpositive number. That means we must have $x > 0$ and $1 - 4x > 0.$ Can you solve the second inequality for $x?$

**skeeter** · June 24th 2012, 02:51 PM

Originally Posted by Maskawisewin

I'm working with: $f(x)=ln(x)-ln(1-4x)$ and I need to figure out the domain of this function. I know that natural logs cannot be zero or negative numbers but is there a trick to simplying this to seeing the domain easily? Thanks.

note that both $x > 0$ and $1-4x > 0$

since $1-4x > 0 \implies x < \frac{1}{4}$ , the intersection of these two restrictions is $0 < x < \frac{1}{4}$ ... the domain.

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