Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
log Subscript b Baseline left parenthesis y Superscript 6 Baseline x right parenthesis
Question content area bottom
Part 1
log Subscript b Baseline left parenthesis y Superscript 6 Baseline x right parenthesis equals nothing

Answer 1

To expand the logarithmic expression \(\log_b(y^6 x)\) using properties of logarithms, we apply the following rules:

1. **Product Rule:** \(\log_b(MN) = \log_b(M) + \log_b(N)\)
2. **Power Rule:** \(\log_b(M^k) = k \log_b(M)\)

Using these properties, we can expand \(\log_b(y^6 x)\) step by step:

1. **Apply the Product Rule:**
\[
\log_b(y^6 x) = \log_b(y^6) + \log_b(x)
\]

2. **Apply the Power Rule to \(\log_b(y^6)\):**
\[
\log_b(y^6) = 6 \log_b(y)
\]

Now, substituting this back into the expression gives us:
\[
\log_b(y^6 x) = 6 \log_b(y) + \log_b(x)
\]

Thus, the fully expanded form of \(\log_b(y^6 x)\) is:
\[
\log_b(y^6 x) = 6 \log_b(y) + \log_b(x)
\]

### Final Expanded Expression:
\[
\log_b(y^6 x) = 6 \log_b(y) + \log_b(x)
\]