Question

use the change of base formula and a calculator to approximate \\(\log_{7}0.9\\)\\(\log_{7}0.9\approx\square\\)(do not round until the final answer. then round to four decimal

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 $c>0,c
eq1$). We can use $c = 10$ (common logarithm) or $c=e$ (natural logarithm). Let's use common logarithm here. So for $\log_{7}0.9$, we have $\log_{7}0.9=\frac{\log 0.9}{\log 7}$.

Step2: Calculate numerator and denominator

First, calculate $\log 0.9$ (base 10 logarithm of 0.9) and $\log 7$ (base 10 logarithm of 7) using a calculator.
$\log 0.9\approx - 0.04575749056$, $\log 7\approx0.84509804$.

Step3: Divide the two values

Now, divide the numerator by the denominator: $\frac{- 0.04575749056}{0.84509804}\approx - 0.054145$.

Step4: Round to four decimal places

Rounding $- 0.054145$ to four decimal places gives $-0.0541$.

Answer:

$\log_{7}0.9\approx\boxed{-0.0541}$