Find the numerical value of the log expression. {:[log a=2quad log b=5quad log c=-8],[log ((b^(5))/(sqrt(a^(3)c^(5))))]:} Answer:

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Find the numerical value of the log expression.  {:[log a=2quad log b=5quad log c=-8],[log ((b^(5))/(sqrt(a^(3)c^(5))))]:} Answer:

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Apply Quotient Rule: Let's start by expressing the given logarithm using the properties of logarithms. We have the expression log((b^(5))/(sqrt(a^(3)c^(5)))). Using the quotient rule of logarithms, which states that log(a/b) = log(a) - log(b), we can separate the numerator and the denominator.Convert Square Root to Exponent: Now, we focus on the denominator, which contains a square root. The square root can be expressed as an exponent of 1/2. So, sqrt(a^(3)c^(5)) can be written as (a^(3)c^(5))^(1/2). Using the power rule of logarithms, which states that log(a^n) = n*log(a), we can take the exponent outside the logarithm.Apply Power Rule: Applying the power rule to both the numerator and the denominator, we get: log(b^(5)) - log((a^(3)c^(5))^(1/2)). This simplifies to 5*log(b) - (1/2)*log(a^(3)c^(5)).Apply Product Rule: Next, we apply the product rule of logarithms, which states that log(a*b) = log(a) + log(b), to the term log(a^(3)c^(5)). This gives us: (1/2)*(log(a^(3)) + log(c^(5))).Apply Power Rule Again: Now, we apply the power rule again to the terms log(a^(3)) and log(c^(5)), which gives us: (1/2)*(3*log(a) + 5*log(c)).Substitute Given Values: Substituting the given values of log a, log b, and log c into the expression, we get: 5*log(b) - (1/2)*(3*log(a) + 5*log(c)) becomes 5*5 - (1/2)*(3*2 + 5*(-8)).Perform Arithmetic Operations: Performing the arithmetic operations, we have: 25 - (1/2)*(6 - 40) which simplifies to 25 - (1/2)*(-34).Calculate Final Value: Finally, we calculate the value of the expression: 25 - (1/2)*(-34) = 25 - (-17) = 25 + 17 = 42.

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=2 \quad \log b=5 \quad \log c=-8 \\ \log \frac{b^{5}}{\sqrt{a^{3} c^{5}}} \end{array}$ $\newline$ Answer:

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Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=2 \quad \log b=5 \quad \log c=-8 \\ \log \frac{b^{5}}{\sqrt{a^{3} c^{5}}} \end{array}$ $\newline$ Answer: