Question

if \\(\log_{b}x = 0.32\\) and \\(\log_{b}z = 0.83\\), evaluate the following expression.\\(\log_{b}\frac{\sqrt{x}}{\sqrt3{z}}\\)\\(\log_{b}\frac{\sqrt{x}}{\sqrt3{z}} \approx \square\\)\\((\text{type an integer or decimal rounded to four decimal places as needed.})\\)

Explanation:

Step1: Apply log quotient rule

The logarithm of a quotient is the difference of the logarithms, so $\log_b\frac{\sqrt{x}}{\sqrt[3]{z}}=\log_b\sqrt{x}-\log_b\sqrt[3]{z}$.

Step2: Apply log power rule

The logarithm of a power is the product of the exponent and the logarithm, so $\log_b\sqrt{x}=\log_bx^{\frac{1}{2}}=\frac{1}{2}\log_bx$ and $\log_b\sqrt[3]{z}=\log_bz^{\frac{1}{3}}=\frac{1}{3}\log_bz$.

Step3: Substitute given values

We know $\log_bx = 0.32$ and $\log_bz = 0.83$. Substitute these into the expressions: $\frac{1}{2}\times0.32-\frac{1}{3}\times0.83$.

Step4: Calculate each term

$\frac{1}{2}\times0.32 = 0.16$ and $\frac{1}{3}\times0.83\approx0.276667$.

Step5: Subtract the terms

$0.16 - 0.276667=- 0.116667\approx - 0.1167$.

Answer:

\(-0.1167\)