Question

What is lnx+2lny−lnz written as a single logarithm?

Answer 1

Answer:

$ln(\frac{xy^{2}}{z})$

Step-by-step explanation:

3 rules of logarithms that we are going to use on this is:

1. $ln(a+b)=ln(a*b)$
2. $ln(a-b)=ln(\frac{a}{b})$
3. $bln(a)=ln(a)^{b}$

We can use the third rule to simplify the expression as:

$ln(x)+2ln(y)-ln(z)\\=ln(x)+ln(y^{2})-ln(z)$

Simplifying the first 2 terms using first rule gives us:

$ln(x*y^{2})-ln(z)$

Now using the second rule to further simplify and write as single logarithm gives us:

$ln(\frac{xy^{2}}{z})$

Answer 2

$\ln x+2\ln y-\ln z=\ln x+\ln y^2-\ln z=\ln\dfrac{xy^2}{z}\\\\Used:\ln a+\ln b=\ln(a\cdot b}\\\\\ln a-\ln b=\ln\dfrac{a}{b}\\\\n\ln a=\ln a^n$