use properties of logarithms to expand the logarithmic expression as much as possible. evaluate logarithmic…

use properties of logarithms to expand the logarithmic expression as much as possible. evaluate logarithmic expressions without using a calculator if possible. \n\\ln \\left( \\frac{e^7}{17} \\right)\n\\ln \\left( \\frac{e^7}{17} \\right) = \\square

use properties of logarithms to expand the logarithmic expression as much as possible. evaluate logarithmic expressions without using a calculator if possible. \n\\ln \\left( \\frac{e^7}{17} \\right)\n\\ln \\left( \\frac{e^7}{17} \\right) = \\square

Answer

Explanation:

Step1: Apply Quotient Rule

The quotient rule for logarithms states that $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$. So, for $\ln\left(\frac{e^7}{17}\right)$, we have $\ln(e^7) - \ln(17)$.

Step2: Apply Power Rule

The power rule for logarithms states that $\ln(a^n) = n\ln(a)$. For $\ln(e^7)$, this becomes $7\ln(e)$.

Step3: Evaluate $\ln(e)$

We know that $\ln(e) = 1$ (since the natural logarithm of its base $e$ is 1). So, $7\ln(e) = 7\times1 = 7$.

Step4: Combine Terms

Putting it all together, $\ln\left(\frac{e^7}{17}\right) = 7 - \ln(17)$.

Answer:

$7 - \ln(17)$