Question

assume $\log_{b}x = 0.34$, $\log_{b}y = 0.52$, and $\log_{b}z = 0.83$. evaluate.\\(\log_{b}\dfrac{\sqrt{xy}}{z}\\)\\(\log_{b}\dfrac{\sqrt{xy}}{z}=\square\\)\\((\text{type an integer or a decimal.})\\)

Explanation:

Step1: Use log quotient rule

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

Step2: Use log power rule on $\sqrt{xy}$

The square root is the same as the 1/2 power, so $\log_b \sqrt{xy} = \log_b (xy)^{\frac{1}{2}} = \frac{1}{2}\log_b (xy)$ by the power rule of logarithms.

Step3: Use log product rule on $\log_b (xy)$

The logarithm of a product is the sum of the logarithms, so $\frac{1}{2}\log_b (xy) = \frac{1}{2}(\log_b x + \log_b y)$.

Step4: Substitute known values

We know $\log_b x = 0.34$, $\log_b y = 0.52$, and $\log_b z = 0.83$. Substitute these into the expression:
First, calculate $\frac{1}{2}(\log_b x + \log_b y) = \frac{1}{2}(0.34 + 0.52) = \frac{1}{2}(0.86) = 0.43$.
Then, subtract $\log_b z$: $0.43 - 0.83 = -0.4$.

Answer:

-0.4