2.12 Properties of Logarithms check

Front

What is the property of logarithms that allows you to expand the expression \( a \ln b \)?

Back

The property states that \( a \ln b = \ln(b^a) \).

Front

Expand the expression: \( 4 \ln(3y) - 3 \ln(x) \)

Back

\( \ln(3) + 4 \ln(y) - 3 \ln(x) \)

Front

What is the logarithmic identity for \( \log(a) + \log(b) \)?

Back

The identity states that \( \log(a) + \log(b) = \log(ab) \).

Front

What is the logarithmic identity for \( \log(a) - \log(b) \)?

Back

The identity states that \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \).

Front

If \( \log(3) = a \) and \( \log(4) = b \), how can you express \( \log(12) \)?

Back

\( \log(12) = a + b \) because \( 12 = 3 \times 4 \).

Front

If \( \log(3) = a \) and \( \log(4) = b \), how can you express \( \log\left(\sqrt[3]{36}\right) \)?

Back

\( \log\left(\sqrt[3]{36}\right) = \frac{2}{3}a + \frac{1}{3}b \) because \( 36 = 4 \times 9 = 4 \times 3^2 \).

Front

If \( \log(3) = a \) and \( \log(4) = b \), how can you express \( \log\left(\frac{16}{27}\right) \)?

Back

\( \log\left(\frac{16}{27}\right) = 2b - 3a \) because \( 16 = 4^2 \) and \( 27 = 3^3 \).

Front

What is the change of base formula for logarithms?

Back

The change of base formula states that \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \) for any positive base \( k \).

Front

What does the logarithm of 1 equal to?

Back

The logarithm of 1 equals 0, i.e., \( \log_b(1) = 0 \) for any base \( b \).

Front

What is the relationship between logarithms and exponents?

Back

Logarithms are the inverse operations of exponentiation, i.e., if \( b^y = x \), then \( \log_b(x) = y \).