Question

what value is equivalent to \\(\log_{4} 2.1\\) to the nearest thousandth? use the change of base formula to determine your answer. (1 point)\\(\bigcirc\\) 1.868\\(\bigcirc\\) 0.535\\(\bigcirc\\) 0.081\\(\bigcirc\\) 0.287

Explanation:

Step1: Recall Change of Base Formula

The change of base formula for logarithms is $\log_b a=\frac{\log_c a}{\log_c b}$, where $c$ can be any positive number not equal to 1. We can use $c = 10$ (common logarithm) or $c=e$ (natural logarithm). Let's use common logarithm (base 10) here. So, for $\log_4 2.1$, we have $\log_4 2.1=\frac{\log_{10} 2.1}{\log_{10} 4}$.

Step2: Calculate Numerator and Denominator

First, calculate $\log_{10} 2.1$. Using a calculator, $\log_{10} 2.1\approx0.3222$.
Next, calculate $\log_{10} 4$. We know that $\log_{10} 4=\log_{10} 2^2 = 2\log_{10} 2\approx2\times0.3010 = 0.6020$ (or directly calculate $\log_{10} 4\approx0.6021$ using a calculator).

Step3: Divide the Two Values

Now, divide the numerator by the denominator: $\frac{0.3222}{0.6021}\approx0.535$.

Answer:

0.535