Question

Log 3√b b/3√a+3/log 3 √ab (a√b) +2loga √b

Answer 1

$log_{b}a=2\\\\\log_{\sqrt[3]{b}}\frac{b}{\sqrt[3]{a}}- \frac{3}{\log_{\sqrt[3]{ab}}(a\sqrt{b})} +2\log_{a}\sqrt{b}=\\\\=3\log_{b}\left (b\cdot a^{-\frac{1}{3}}\right )-\frac{3}{3\log_{ab}(a\cdot b^{\frac{1}{2}})}+2\log_{a}b^{\frac{1}{2}}=\\\\=3\Big (1-\frac{1}{3}\log_{b}a\Big )- \frac{\log_{b}(ab)}{\log_{b}(a\cdot b^{\frac{1}{2}})} +2\cdot \frac{1}{2}\cdot \log_{a}b=\\\\=3-\log_{b}a- \frac{\log_{b}a+1}{\log_{b}a+\frac{1}{2}\log_{b}b} + \frac{1}{\log_{b}a} =$

$=3-2- \frac{2+1}{2+\frac{1}{2}} +\frac{1}{2}=\frac{3}{2}-\frac{3}{5/2}=\frac{3}{2}-\frac{6}{5}=\frac{15-12}{10}=\frac{3}{10}=0,3$

Answer 2

To simplify the expression, let's break it down step by step:

1. Logarithmic properties:

   • Logₓ(a) + Logₓ(b) = Logₓ(ab)
   • Logₓ(a) - Logₓ(b) = Logₓ(a/b)
   • Logₓ(a^m) = m * Logₓ(a)

2. Simplify each term: a. Log₃(√b) Since the square root can be expressed as a fractional exponent, we can rewrite it as: = Log₃(b^(1/2)) Using the property Logₓ(a^m) = m * Logₓ(a): = (1/2) * Log₃(b)

   b. b/(3√a) We can rewrite the cube root as a fractional exponent: = b/(a^(1/3)) Using the property Logₓ(a/b) = Logₓ(a) - Logₓ(b): = Logₐ(a) - Logₐ(b) = 1/3 * Logₐ(a) - Logₐ(b)

   c. 3/log₃(√ab(a√b)) + 2 * Logₐ(√b) Simplifying the square root: = 3/log₃(ab^(1/2)(a^(1/2)b^(1/2))) Using the property Logₓ(a^m) = m * Logₓ(a): = 3/log₃(ab^(1/2) * a^(1/2) * b^(1/2)) Applying the logarithmic property Logₓ(a) + Logₓ(b) = Logₓ(ab): = 3/log₃((ab)^(1/2) * (ab)^(1/2)) Simplifying: = 3/log₃(ab * ab) = 3/log₃(a^2b^2) = 3/(2 * Log₃(a) + 2 * Log₃(b))

Now, putting it all together: Log₃√b b/3√a + 3/Log₃√ab (a√b) + 2 * Logₐ√b = (1/2) * Log₃(b) + 1/3 * Logₐ(a) - Logₐ(b) + 3/(2 * Log₃(a) + 2 * Log₃(b)) + 2 * Logₐ(√b)