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Find the numerical value of the log expression.

{:[log a=8quad log b=2quad log c=-4],[log ((c^(8))/(a^(9)b^(9)))]:}
Answer:

Find the numerical value of the log expression.\newlineloga=8logb=2logc=4logc8a9b9 \begin{array}{c} \log a=8 \quad \log b=2 \quad \log c=-4 \\ \log \frac{c^{8}}{a^{9} b^{9}} \end{array} \newlineAnswer:

Full solution

Q. Find the numerical value of the log expression.\newlineloga=8logb=2logc=4logc8a9b9 \begin{array}{c} \log a=8 \quad \log b=2 \quad \log c=-4 \\ \log \frac{c^{8}}{a^{9} b^{9}} \end{array} \newlineAnswer:
  1. Apply Quotient Rule: We are given the values of loga\log a, logb\log b, and logc\log c, and we need to find the value of log(c8a9b9)\log\left(\frac{c^{8}}{a^{9}b^{9}}\right). We can use the properties of logarithms to simplify the expression.\newlineFirst, let's apply the quotient rule of logarithms, which states that log(ab)=log(a)log(b)\log\left(\frac{a}{b}\right) = \log(a) - \log(b).\newlinelog(c8a9b9)=log(c8)log(a9b9)\log\left(\frac{c^{8}}{a^{9}b^{9}}\right) = \log(c^{8}) - \log(a^{9}b^{9})
  2. Apply Product Rule: Now, let's apply the product rule of logarithms to the second term, which states that log(ab)=log(a)+log(b)\log(ab) = \log(a) + \log(b). \newlinelog(a9b9)=log(a9)+log(b9)\log(a^{9}b^{9}) = \log(a^{9}) + \log(b^{9})
  3. Apply Power Rule: Next, we apply the power rule of logarithms, which states that log(an)=nlog(a)\log(a^n) = n\log(a), to each term.\newlinelog(c8)=8log(c)\log(c^{8}) = 8\log(c)\newlinelog(a9)=9log(a)\log(a^{9}) = 9\log(a)\newlinelog(b9)=9log(b)\log(b^{9}) = 9\log(b)
  4. Substitute Given Values: Now we substitute the given values of loga\log a, logb\log b, and logc\log c into the expression.\newlinelog(c8)(log(a9)+log(b9))=8log(c)(9log(a)+9log(b))\log(c^{8}) - (\log(a^{9}) + \log(b^{9})) = 8\cdot\log(c) - (9\cdot\log(a) + 9\cdot\log(b))\newline=8(4)(98+92)= 8\cdot(-4) - (9\cdot8 + 9\cdot2)
  5. Perform Arithmetic Operations: We perform the arithmetic operations.\newline=32(72+18)= -32 - (72 + 18)\newline=3290= -32 - 90\newline=122= -122

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