Simplify log 6 - log 15 + 2 log 5 and log (5/8) - 2 log (5/9) + log 40

1(a)

Step 1: Start with the given equation.
$\log b = \log 15 + 2 \log 5$

Step 2: Apply the power rule: $n \log a = \log (a^n)$.
$2 \log 5 = \log (5^2)$
$\log b = \log 15 + \log (5^2)$

Step 3: Apply the product rule: $\log a + \log c = \log (a \cdot c)$.
$\log b = \log (15 \cdot 5^2)$

Step 4: Compute $5^2 = 25$.
$\log b = \log (15 \cdot 25)$

Step 5: Compute $15 \cdot 25 = 375$.
$\log b = \log 375$

Step 6: If $\log b = \log 375$, then $b = 375$.
Final answer: $b = 375$
375

2)

Step 1: Start with the given expression.
$\log \frac{5}{3} - 2 \log \frac{3}{7} + \log 44$

Step 2: Apply the power rule to the second term: $n \log a = \log (a^n)$.
$-2 \log \frac{3}{7} = -\log \left( \frac{3}{7} \right)^2 = -\log \frac{9}{49}$

Step 3: Apply the quotient rule to the negative log: $-\log \frac{9}{49} = \log \frac{49}{9}$.
$\log \frac{5}{3} + \log \frac{49}{9} + \log 44$

Step 4: Apply the product rule for logs: $\log a + \log b + \log c = \log (a \cdot b \cdot c)$.
$\log \left( \frac{5}{3} \cdot \frac{49}{9} \cdot 44 \right)$

Step 5: Simplify the argument step-by-step. First, $\frac{5}{3} \cdot \frac{49}{9}$.
Numerator: $5 \cdot 49 = 245$
Denominator: $3 \cdot 9 = 27$
$\frac{245}{27} \cdot 44$

Step 6: Now multiply by 44.
$245 \cdot 44 = 10780$
$\log \frac{10780}{27}$

Step 7: Check if $\frac{10780}{27}$ simplifies (gcd of 10780 and 27 is 1).
Final answer: $\log \frac{10780}{27}$
\log \dfrac{10780{27}}