Questions: loga b=4, where a>0 and b>0, what is the value of log(a^2)(b^3) ? (A) 2 (B) 8/3 (C) 6 (D) 12

Transcript text: $\log _{a} b=4$, where $a>0$ and $b>0$, what is the value of $\log _{a^{2}}\left(b^{3}\right) ?$ (A) 2 (B) $\frac{8}{3}$ (C) 6 (D) 12

Solution

Solution Steps

Step 1: Understand the given information

We are given that $\log_{a} b = 4$, where $a > 0$ and $b > 0$. This means that $a^4 = b$.

Step 2: Express $\log_{a^{2}}\left(b^{3}\right)$ in terms of $\log_{a} b$

We need to find the value of $\log_{a^{2}}\left(b^{3}\right)$. Using the change of base formula for logarithms, we can rewrite this as: \[ \log_{a^{2}}\left(b^{3}\right) = \frac{\log_{a} \left(b^{3}\right)}{\log_{a} \left(a^{2}\right)} \]

Step 3: Simplify the numerator and denominator

First, simplify the numerator: \[ \log_{a} \left(b^{3}\right) = 3 \log_{a} b = 3 \times 4 = 12 \] Next, simplify the denominator: \[ \log_{a} \left(a^{2}\right) = 2 \log_{a} a = 2 \times 1 = 2 \]

Step 4: Compute the value of $\log_{a^{2}}\left(b^{3}\right)$

Now, substitute the simplified numerator and denominator back into the expression: \[ \log_{a^{2}}\left(b^{3}\right) = \frac{12}{2} = 6 \]