Questions: Write the following as a single logarithm. Assume all variables are positive. 4 ln (a)+6 ln (c)=

Solution Steps

To combine the given logarithmic expressions into a single logarithm, we can use the properties of logarithms. Specifically, we can use the power rule, which states that \( n \ln(x) = \ln(x^n) \), and the product rule, which states that \( \ln(x) + \ln(y) = \ln(xy) \). First, apply the power rule to each term, and then use the product rule to combine them.

The power rule of logarithms states that \( n \ln(x) = \ln(x^n) \). We apply this rule to each term in the expression:

• For \( 4 \ln(a) \), we have \( \ln(a^4) \).
• For \( 6 \ln(c) \), we have \( \ln(c^6) \).

The product rule of logarithms states that \( \ln(x) + \ln(y) = \ln(xy) \). We use this rule to combine the two logarithmic expressions:

• Combine \( \ln(a^4) \) and \( \ln(c^6) \) to get \( \ln(a^4 \cdot c^6) \).

Final Answer