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Write the expression
(1)/(3)ln 8+ln 3 as a single logarithm in simplest form without any negative exponents.
Answer:
ln(◻)

Write the expression $\frac{1}{3} \ln 8+\ln 3$ as a single logarithm in simplest form without any negative exponents. $\newline$ Answer: $\ln (\square)$

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Power rule with rational exponents

Full solution

Q. Write the expression

\frac{1}{3} \ln 8+\ln 3

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

\newline

Answer:

\ln (\square)

Apply Power Rule: We have the expression $(\frac{1}{3})\ln 8 + \ln 3$ . To combine these logarithms into a single logarithm, we can use the logarithm properties.
Simplify $8^{1/3}$ : First, we apply the power rule of logarithms, which states that $a \cdot \ln(b) = \ln(b^a)$ , to the term $(1/3)\ln 8$ . $\newline$ $(1/3)\ln 8 = \ln(8^{1/3})$
Substitute $2$ : Now we simplify $8^{1/3}$ . Since $8$ is $2^3$ , we can write $8^{1/3}$ as $(2^3)^{1/3}$ . $\newline$ $8^{1/3} = (2^3)^{1/3}$
Combine Logarithms: Using the power of a power rule, which states $(a^n)^m = a^{n*m}$ , we simplify $(2^3)^{1/3}$ . $(2^3)^{1/3} = 2^{3*(1/3)} = 2^1 = 2$
Calculate Product: Now we substitute $2$ back into the logarithm to get $\ln(2)$ . $\newline$ $\ln(8^{1/3}) = \ln(2)$
Write Final Expression: We now have $\ln(2) + \ln(3)$ . To combine these two logarithms into a single logarithm, we use the product rule of logarithms, which states $\ln(a) + \ln(b) = \ln(a*b)$ . $\newline$ $\ln(2) + \ln(3) = \ln(2*3)$
Write Final Expression: We now have $\ln(2) + \ln(3)$ . To combine these two logarithms into a single logarithm, we use the product rule of logarithms, which states $\ln(a) + \ln(b) = \ln(a*b)$ . $\ln(2) + \ln(3) = \ln(2*3)$ We calculate the product $2*3$ . $2*3 = 6$
Write Final Expression: We now have $\ln(2) + \ln(3)$ . To combine these two logarithms into a single logarithm, we use the product rule of logarithms, which states $\ln(a) + \ln(b) = \ln(a*b)$ . $\ln(2) + \ln(3) = \ln(2*3)$ We calculate the product $2*3$ . $2*3 = 6$ Finally, we write the expression as a single logarithm. $\ln(2*3) = \ln(6)$

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