Given the substitutions ln 2=a,ln 3=b, and ln 5=c, find the value of ln(81root(3)(e)) in terms of a,b, and c. Answer:

Simplify Expression: We need to express ln(81√3(e)) in terms of a, b, and c. First, let's simplify the expression inside the logarithm. The cube root of e is e^(1/3), and 81 can be written as 3^4. So, ln(81√3(e)) becomes ln(3^4 * e^(1/3)).Apply Logarithm Properties: Now, we apply the properties of logarithms to separate the terms. The logarithm of a product is the sum of the logarithms, so ln(3^4 * e^(1/3)) becomes ln(3^4) + ln(e^(1/3)).Apply Power Rule: Next, we apply the power rule of logarithms, which states that ln(x^y) = y * ln(x). This gives us 4 * ln(3) + (1/3) * ln(e).Substitute Given Values: Since ln(e) is equal to 1, the expression simplifies to 4 * ln(3) + (1/3) * 1, which is 4 * ln(3) + 1/3.Final Expression: Now, we substitute the given values for ln(2), ln(3), and ln(5). Since ln(3) = b, we replace ln(3) with b in our expression. This gives us 4 * b + 1/3.The final expression is 4b + 1/3. This is the value of ln(81√3(e)) in terms of a, b, and c, although a and c do not appear in the final expression.

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Given the substitutions
ln 2=a,ln 3=b, and
ln 5=c, find the value of
ln(81root(3)(e)) in terms of
a,b, and
c.
Answer:

Given the substitutions $\ln 2=a, \ln 3=b$ , and $\ln 5=c$ , find the value of $\ln (81 \sqrt[3]{e})$ in terms of $a, b$ , and $c$ . $\newline$ Answer:

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Q. Given the substitutions

\ln 2=a, \ln 3=b

, and

\ln 5=c

, find the value of

\ln (81 \sqrt[3]{e})

in terms of

a, b

, and

c

\newline

Answer:

Simplify Expression: We need to express $\ln(81\sqrt[3]{e})$ in terms of $a$ , $b$ , and $c$ . First, let's simplify the expression inside the logarithm. The cube root of $e$ is $e^{1/3}$ , and $81$ can be written as $3^4$ . $\newline$ So, $\ln(81\sqrt[3]{e})$ becomes $\ln(3^4 \cdot e^{1/3})$ .
Apply Logarithm Properties: Now, we apply the properties of logarithms to separate the terms. The logarithm of a product is the sum of the logarithms, so $\ln(3^4 \cdot e^{1/3})$ becomes $\ln(3^4) + \ln(e^{1/3})$ .
Apply Power Rule: Next, we apply the power rule of logarithms, which states that $\ln(x^y) = y \cdot \ln(x)$ . This gives us $4 \cdot \ln(3) + \left(\frac{1}{3}\right) \cdot \ln(e)$ .
Substitute Given Values: Since $\ln(e)$ is equal to $1$ , the expression simplifies to $4 \times \ln(3) + \left(\frac{1}{3}\right) \times 1$ , which is $4 \times \ln(3) + \frac{1}{3}$ .
Final Expression: Now, we substitute the given values for $\ln(2)$ , $\ln(3)$ , and $\ln(5)$ . Since $\ln(3) = b$ , we replace $\ln(3)$ with $b$ in our expression. This gives us $4 \times b + \frac{1}{3}$ .
Final Expression: Now, we substitute the given values for $\ln(2)$ , $\ln(3)$ , and $\ln(5)$ . Since $\ln(3) = b$ , we replace $\ln(3)$ with $b$ in our expression. This gives us $4 \cdot b + \frac{1}{3}$ .The final expression is $4b + \frac{1}{3}$ . This is the value of $\ln(81\sqrt{3}(e))$ in terms of $a$ , $b$ , and $\ln(3)$ $1$ , although $a$ and $\ln(3)$ $1$ do not appear in the final expression.