Question

What is \( \ln x + 2\ln y - \ln z \) written as a single logarithm?

a) \( \ln\left(\frac{{y^2z}}{{x}}\right) \)
b) \( \ln\left(\frac{{2xyz}}{{z}}\right) \)
c) \( \ln(xy^2z) \)
d) \( \ln(xy^2z) \)

Answer 1

To write ln x + 2ln y - ln z as a single logarithm, we can simplify it using the properties of logarithms. The expression can be written as ln(xy^2/z).

Step-by-step explanation:

To write ln x + 2ln y - ln z as a single logarithm, we can use the exponent rules for logarithms. From the rule ln xy = ln x + ln y, we can rewrite the expression as ln(x) + ln(y^2) - ln(z). Then, using the rule ln(A/B) = ln A - ln B, we can further simplify it to ln(x) + ln(y^2/z). Finally, by applying the rule ln(x^k) = k ln(x), we can rewrite it as a single logarithm: ln(x) + 2ln(y) - ln(z) = ln(x) + ln(y^2/z) = ln(xy^2/z).