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The image contains 14 logarithmic and exponential expressions. Here’s a breakdown of each equation listed on the board:

1. \( y = \ln(7b^6) \)
2. \( y = \log \sqrt[3]{x^5 - 2x} \)
3. \( y = \ln(6x^4) \)
4. \( y = \ln\sqrt[6]{7x^2 - 2} \)
5. \( y = \log(x^3 - 2x + 3) \)
6. \( y = \ln\left[(9x - 1)^5(6x^2 - 2)^3\right] \)
7. \( y = \log\left(\frac{x+7}{x-7}\right) \)
8. \( y = \ln^7(4x^6) \)
9. \( y = \frac{\sqrt{x}}{\ln(x^2)} \)
10. \( y = 2x^5 \log(5x) \)
11. \( y = \sqrt[3]{\log(8x^2)} \)
12. \( y = \log\left(\frac{5x^3}{x^4}\right) \)
13. \( y = \sqrt{(\ln(a x^3))^5} \)
14. \( y = x^6 \ln(x) \)

These equations involve the use of natural logarithms (\(\ln\)), common logarithms (\(\log\)), and various algebraic manipulations such as exponents and roots. 

Would you like a detailed explanation or solution for any specific equation? 

Here are five related questions to expand your understanding:

1. What are the properties of logarithms that simplify expressions like those in equations 1-14?
2. How do you differentiate an equation involving logarithmic functions?
3. What is the relationship between the natural logarithm (\(\ln\)) and the common logarithm (\(\log\))?
4. How do logarithmic and exponential functions behave asymptotically as \(x\) approaches infinity?
5. How can you solve an equation involving both exponential and logarithmic terms, such as \(x^6 \ln(x) = C\) for some constant \(C\)?

**Tip:** Understanding the properties of logarithms, such as the product, quotient, and power rules, can greatly simplify solving and manipulating complex logarithmic expressions.

The image contains 14 logarithmic and exponential expressions. Can you provide detailed explanations or solutions?

Explore 14 logarithmic and exponential equations involving natural logs, common logarithms, and various algebraic manipulations. Learn key logarithmic properties and exponential function rules, suitable for high school students in Grades 10-12.

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