Full solution

Q. Find the numerical value of the log expression.

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\begin{array}{c} \log a=8 \quad \log b=1 \quad \log c=-4 \\ \log \frac{a^{5} b^{4}}{\sqrt{c^{3}}} \end{array}

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Answer:

Apply Quotient Rule: We are given the values of $\log a$ , $\log b$ , and $\log c$ , and we need to find the value of $\log\left(\frac{a^{5}b^{4}}{\sqrt{c^{3}}}\right)$ . Let's start by applying the quotient rule of logarithms, which states that $\log\left(\frac{x}{y}\right) = \log(x) - \log(y)$ .
Deal with Numerator: Now, we need to deal with the numerator and the denominator separately. For the numerator, which is $a^{5}b^{4}$ , we can apply the product rule of logarithms, which states that $\log(xy) = \log(x) + \log(y)$ . We also need to consider the exponents, using the power rule of logarithms, which states that $\log(x^{k}) = k\cdot\log(x)$ .
Deal with Denominator: Applying the product and power rules to the numerator, we get $\log(a^{5}) + \log(b^{4})$ which simplifies to $5\log(a) + 4\log(b)$ . Substituting the given values, we have $5\times 8 + 4\times 1$ , which equals $40 + 4 = 44$ .
Combine Results: For the denominator, we have $\sqrt{c^{3}}$ , which can be rewritten as $c^{\frac{3}{2}}$ . Applying the power rule of logarithms, we get $\left(\frac{3}{2}\right)\log(c)$ . Substituting the given value of $\log c$ , we have $\left(\frac{3}{2}\right)(-4)$ , which equals $-6$ .
Combine Results: For the denominator, we have $\sqrt{c^{3}}$ , which can be rewritten as $c^{\frac{3}{2}}$ . Applying the power rule of logarithms, we get $(\frac{3}{2})\log(c)$ . Substituting the given value of $\log c$ , we have $(\frac{3}{2})(-4)$ , which equals $-6$ .Now we can combine the results for the numerator and the denominator using the quotient rule we applied in the first step. We have $\log(\frac{a^{5}b^{4}}{\sqrt{c^{3}}}) = \log(a^{5}b^{4}) - \log(\sqrt{c^{3}})$ which simplifies to $44 - (-6)$ , giving us a final result of $44 + 6 = 50$ .