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Express the given expression as an integer or as a fraction in simplest form. (6^(log_(6)17+log_(6)3)) Answer:

Express the given expression as an integer or as a fraction in simplest form.\newline(6log⁡617+log⁡63) \left(6^{\log _{6} 17+\log _{6} 3}\right) \newlineAnswer:

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Full solution

Q. Express the given expression as an integer or as a fraction in simplest form.\newline(6log⁡617+log⁡63) \left(6^{\log _{6} 17+\log _{6} 3}\right) \newlineAnswer:
  1. Combine logarithms: Apply the property of logarithms that states log⁡b(m)+log⁡b(n)=log⁡b(m∗n)\log_b(m) + \log_b(n) = \log_b(m*n) to combine the logarithms.log⁡617+log⁡63=log⁡6(17∗3)\log_{6}17 + \log_{6}3 = \log_{6}(17*3)
  2. Calculate product: Calculate the product inside the logarithm. 17×3=5117 \times 3 = 51
  3. Substitute back: Substitute the combined logarithm with the product back into the original expression.\newline6log⁡617+log⁡636^{\log_{6}17+\log_{6}3} = 6log⁡6516^{\log_{6}51}
  4. Apply property: Apply the property of logarithms that states blog⁡b(m)=mb^{\log_b(m)} = m, where bb is the base of the logarithm and mm is the argument.\newline6log⁡651=516^{\log_{6}51} = 51
  5. Simplify result: Since 5151 is already an integer, it is in its simplest form.

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