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Express the given expression as an integer or as a fraction in simplest form. (4^(log_(4)10-log_(4)4)) Answer:

Express the given expression as an integer or as a fraction in simplest form.\newline(4log⁡410−log⁡44) \left(4^{\log _{4} 10-\log _{4} 4}\right) \newlineAnswer:

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Full solution

Q. Express the given expression as an integer or as a fraction in simplest form.\newline(4log⁡410−log⁡44) \left(4^{\log _{4} 10-\log _{4} 4}\right) \newlineAnswer:
  1. Apply Logarithm Properties: Apply the properties of logarithms to simplify the expression inside the exponent. \newlinelog⁡410−log⁡44=log⁡4(104)\log_{4}10 - \log_{4}4 = \log_{4}(\frac{10}{4}) because log⁡b(m)−log⁡b(n)=log⁡b(mn)\log_{b}(m) - \log_{b}(n) = \log_{b}(\frac{m}{n}).
  2. Cancel Base of Exponent and Logarithm: Now we have 4log⁡4(104)4^{\log_{4}(\frac{10}{4})}. Since the base of the exponent and the base of the logarithm are the same, they cancel each other out.\newlineThis means 4log⁡4(104)=1044^{\log_{4}(\frac{10}{4})} = \frac{10}{4}.
  3. Simplify Fraction: Simplify the fraction 104\frac{10}{4} by dividing both numerator and denominator by their greatest common divisor, which is 22.\newline104=(10/2)(4/2)=52.\frac{10}{4} = \frac{(10/2)}{(4/2)} = \frac{5}{2}.
  4. Check Final Result: Check the final result to ensure it is in simplest form. 52\frac{5}{2} cannot be simplified further, so it is already in simplest form.

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