Let's solve the given logarithmic expressions step by step.

Part (a)

The expression is: $\ln\left(\frac{2}{x}\right) + \ln(5x) - \ln(x^3)$

Step 1: Apply the logarithmic property $\ln(a) + \ln(b) = \ln(a \cdot b)$:

$\ln\left(\frac{2}{x}\right) + \ln(5x) = \ln\left(\frac{2}{x} \cdot 5x\right)$

Simplify the product: $\ln\left(\frac{2}{x} \cdot 5x\right) = \ln(10)$

So, the expression becomes: $\ln(10) - \ln(x^3)$

Step 2: Apply the property $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$:

$\ln(10) - \ln(x^3) = \ln\left(\frac{10}{x^3}\right)$

Final simplified result: $\ln\left(\frac{10}{x^3}\right)$

Part (b)

The expression is: $\log(x^4) + \log\left(\frac{3}{x}\right) - \log(x^2) + \log(2)$

Step 1: Apply the power rule $\log(a^b) = b\log(a)$:

$\log(x^4) = 4\log(x), \quad \log(x^2) = 2\log(x)$

So, the expression becomes: $4\log(x) + \log\left(\frac{3}{x}\right) - 2\log(x) + \log(2)$

Step 2: Simplify $\log\left(\frac{3}{x}\right)$ using the property $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$:

$\log\left(\frac{3}{x}\right) = \log(3) - \log(x)$

Substitute into the expression: $4\log(x) + \log(3) - \log(x) - 2\log(x) + \log(2)$

Step 3: Combine like terms:

Combine the $\log(x)$ terms: $4\log(x) - \log(x) - 2\log(x) = \log(x)$

So, the expression becomes: $\log(x) + \log(3) + \log(2)$

Step 4: Apply the property $\log(a) + \log(b) = \log(a \cdot b)$:

$\log(x) + \log(3) + \log(2) = \log\left(x \cdot 3 \cdot 2\right)$

Simplify: $\log\left(6x\right)$

Final simplified result: $\log(6x)$

Final Answers:

(a) $\ln\left(\frac{10}{x^3}\right)$
(b) $\log(6x)$

Let me know if you have further questions or need additional clarification!

Related Questions:

1. What are the general rules of logarithms, and how do they apply to simplification?
2. How does the base of a logarithm affect its calculation?
3. Can logarithmic expressions be simplified differently if additional constraints are applied?
4. What is the importance of the power rule in logarithms?
5. How would these simplifications change if we had base-10 logarithms throughout?

Tip:

Always check for opportunities to combine terms using properties like product, quotient, and power rules for logarithms—they are powerful tools for simplification.