If a=ln2 and b=ln5, find in terms of a and b: (1) ln50 (2) ln0.16

Given \( a = \ln 2 \) and \( b = \ln 5 \), we can express \( \ln 50 \) and \( \ln 0.16 \) in terms of \( a \) and \( b \) as follows:

### 1. \( \ln 50 \)

We can write 50 as:
\[
50 = 2 \times 5^2
\]

So, the natural logarithm of 50 is:
\[
\ln 50 = \ln(2 \times 5^2)
\]

Using the logarithm properties:
\[
\ln(AB) = \ln A + \ln B \quad \text{and} \quad \ln(A^n) = n\ln A
\]
We get:
\[
\ln 50 = \ln 2 + \ln 5^2 = \ln 2 + 2\ln 5
\]

Substituting the given values \( a = \ln 2 \) and \( b = \ln 5 \):
\[
\ln 50 = a + 2b
\]

### 2. \( \ln 0.16 \)

We can write 0.16 as:
\[
0.16 = \frac{16}{100} = \frac{2^4}{10^2} = \frac{2^4}{(2 \times 5)^2} = \frac{2^4}{2^2 \times 5^2} = \frac{2^2}{5^2}
\]

So, the natural logarithm of 0.16 is:
\[
\ln 0.16 = \ln \left(\frac{2^2}{5^2}\right)
\]

Using the logarithm properties:
\[
\ln\left(\frac{A}{B}\right) = \ln A - \ln B
\]
We get:
\[
\ln 0.16 = \ln(2^2) - \ln(5^2) = 2\ln 2 - 2\ln 5
\]

Substituting the given values \( a = \ln 2 \) and \( b = \ln 5 \):
\[
\ln 0.16 = 2a - 2b
\]

### Final Answers:
1. \( \ln 50 = a + 2b \)
2. \( \ln 0.16 = 2a - 2b \)

Would you like more details or have any questions?

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Here are 5 related questions to explore:
1. How would you express \( \ln 1000 \) in terms of \( a \) and \( b \)?
2. What is the value of \( \ln 10 \) using \( a = \ln 2 \) and \( b = \ln 5 \)?
3. If \( c = \ln 3 \), how would you express \( \ln 6 \) in terms of \( a \), \( b \), and \( c \)?
4. Can you express \( \ln 0.2 \) in terms of \( a \) and \( b \)?
5. How would you express \( \ln 25 \) in terms of \( b \)?

**Tip:** Always remember to utilize logarithm properties like product, quotient, and power rules when simplifying logarithmic expressions.

Learn how to express ln 50 and ln 0.16 in terms of ln 2 and ln 5. This problem covers properties of logarithms, including product, quotient, and power rules. Suitable for advanced high school students and above.

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