How can I simplify the expression 12ln(z) + ln(5) + 4ln(y) into a single logarithmic form?

To simplify the expression 12ln(z) + ln(5) + 4ln(y) into a single logarithm, we will use the properties of logarithms.

• Property 1: n * ln(a) = ln(a^n) – This property allows us to bring coefficients in front of a logarithm as exponents.
• Property 2: ln(a) + ln(b) = ln(a * b) – This property allows us to combine logarithms that are added together.

Following these properties, we can start simplifying:

1. Apply the first property to 12ln(z):
   12ln(z) = ln(z^{12})
2. Apply the first property to 4ln(y):
   4ln(y) = ln(y^{4})
3. Now substitute these into the original expression:
   ln(z^{12}) + ln(5) + ln(y^{4})
4. Next, apply the second property to combine the logarithms:
   ln(z^{12} * 5 * y^{4})

So, the simplified expression in a single logarithmic form is:

ln(5 * z^{12} * y^{4})

Thus, the final answer is:
ln(5z¹²y⁴)