Question

use common logarithms or natural logarithms and a calculator to evaluate the expression.
\\(\log_{18} 19\\)

evaluate the expression.
\\(\log_{18} 19 \approx \square\\)
(type an integer or a decimal. do not round until the final answer. then round to four decimal places as needed.)

Explanation:

Step1: Apply Change of Base Formula

The change of base formula for logarithms is $\log_{b}a = \frac{\log a}{\log b}$ (or $\frac{\ln a}{\ln b}$). For $\log_{18}19$, we can use common logarithms (base 10) or natural logarithms (base $e$). Let's use natural logarithms: $\log_{18}19=\frac{\ln 19}{\ln 18}$.

Step2: Calculate the Logarithms

Using a calculator, $\ln 19\approx2.944438979$, $\ln 18\approx2.890371758$.

Step3: Divide the Values

Divide $\ln 19$ by $\ln 18$: $\frac{2.944438979}{2.890371758}\approx1.0187$.

Answer:

$1.0187$