Find the numerical value of the log expression. {:[log a=11quad log b=-7quad log c=-12],[log ((root(3)(c))/(a^(9)b^(6)))]:} Answer:

Question

Find the numerical value of the log expression.  {:[log a=11quad log b=-7quad log c=-12],[log ((root(3)(c))/(a^(9)b^(6)))]:} Answer:

Bytelearn · Accepted Answer

Given Log Values: We are given the values of $ \log a = 11 $, $ \log b = -7 $, and $ \log c = -12 $. We need to find the value of $ \log \left( \frac{\sqrt[3]{c}}{a^{9}b^{6}} \right) $. First, let's apply the logarithm rules to simplify the expression. Using the quotient rule of logarithms, which states that $ \log \left( \frac{x}{y} \right) = \log x - \log y $, we can write: $ \log \left( \frac{\sqrt[3]{c}}{a^{9}b^{6}} \right) = \log \left( \sqrt[3]{c} \right) - \log \left( a^{9}b^{6} \right) $.Simplify Expression: Next, we apply the power rule of logarithms, which states that $ \log \left( x^{n} \right) = n \cdot \log x $, to the terms $ a^{9} $ and $ b^{6} $: $ \log \left( a^{9} \right) = 9 \cdot \log a $ and $ \log \left( b^{6} \right) = 6 \cdot \log b $. We also apply the same rule to $ \sqrt[3]{c} $, which can be written as $ c^{1/3} $, so $ \log \left( \sqrt[3]{c} \right) = \frac{1}{3} \cdot \log c $.Apply Log Rules: Now, we substitute the given logarithm values into the expression: $ \log \left( \sqrt[3]{c} \right) - \log \left( a^{9}b^{6} \right) $ becomes $ \frac{1}{3} \cdot \log c - (9 \cdot \log a + 6 \cdot \log b) $. Substituting the values, we get $ \frac{1}{3} \cdot (-12) - (9 \cdot 11 + 6 \cdot (-7)) $.Substitute Values: Let's perform the calculations: $ \frac{1}{3} \cdot (-12) = -4 $, $ 9 \cdot 11 = 99 $, and $ 6 \cdot (-7) = -42 $. So, the expression becomes $ -4 - (99 - 42) $.Perform Calculations: Finally, we calculate the remaining operation: $ -4 - (99 - 42) = -4 - 99 + 42 = -4 - 57 = -61 $. Therefore, the numerical value of the log expression is $ -61 $.

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=11 \quad \log b=-7 \quad \log c=-12 \\ \log \frac{\sqrt[3]{c}}{a^{9} b^{6}} \end{array}$ $\newline$ Answer:

Full solution

More problems from Quotient property of logarithms

Related topics

Find the numerical value of the log expression.\newlinelog⁡a=11log⁡b=−7log⁡c=−12log⁡c3a9b6 \begin{array}{c} \log a=11 \quad \log b=-7 \quad \log c=-12 \\ \log \frac{\sqrt[3]{c}}{a^{9} b^{6}} \end{array} loga=11logb=−7logc=−12loga9b63c​​​\newlineAnswer:

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=11 \quad \log b=-7 \quad \log c=-12 \\ \log \frac{\sqrt[3]{c}}{a^{9} b^{6}} \end{array}$ $\newline$ Answer: