Full solution

Q. Write the expression

-\ln 2+3

as a single logarithm in simplest form without any negative exponents.

\newline

Answer:

\ln (\square)

Properties of Logarithms: Understand the properties of logarithms. $\newline$ The properties of logarithms that are relevant to this problem are: $\newline$ $1$ . The product rule: $\ln(a) + \ln(b) = \ln(a \cdot b)$ $\newline$ $2$ . The power rule: $\ln(a^b) = b \cdot \ln(a)$ $\newline$ $3$ . The quotient rule: $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ $\newline$ We will use these properties to combine the terms into a single logarithm.
Power Rule Application: Apply the power rule to the term $3$ to express it as a logarithm. $\newline$ Since $3$ is not a logarithm, we need to express it as an equivalent logarithmic expression. We can use the fact that $e^3$ is the number whose natural logarithm is $3$ . Therefore, we can write $3$ as $\ln(e^3)$ . $\newline$ So, $-\ln 2 + 3$ becomes $-\ln 2 + \ln(e^3)$ .
Quotient Rule Application: Apply the quotient rule to combine the logarithms. $\newline$ Using the quotient rule $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$ , we can combine $-\ln 2 + \ln(e^3)$ into a single logarithm. $\newline$ So, $-\ln 2 + \ln(e^3)$ becomes $\ln\left(\frac{e^3}{2}\right)$ .