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Exponential and Logarithmic Functions Writing an expression as a single logarithm Write the expression as a single logarithm. 7log_(c)(w-7)-3log_(c)(w+2)

Exponential and Logarithmic Functions Writing an expression as a single logarithm\newlineWrite the expression as a single logarithm.\newline7log⁡c(w−7)−3log⁡c(w+2) 7 \log _{c}(w-7)-3 \log _{c}(w+2)

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Q. Exponential and Logarithmic Functions Writing an expression as a single logarithm\newlineWrite the expression as a single logarithm.\newline7log⁡c(w−7)−3log⁡c(w+2) 7 \log _{c}(w-7)-3 \log _{c}(w+2)
  1. Apply Power Rule: Apply the power rule of logarithms to both terms.\newlineThe power rule states that log⁡b(an)=nlog⁡b(a)\log_b(a^n) = n\log_b(a). We can apply this rule in reverse to move the coefficients of the logarithms up as exponents of their arguments.\newline7log⁡c(w−7)7\log_c(w-7) becomes log⁡c((w−7)7)\log_c((w-7)^7)\newline3log⁡c(w+2)3\log_c(w+2) becomes log⁡c((w+2)3)\log_c((w+2)^3)
  2. Apply Quotient Rule: Apply the quotient rule of logarithms to combine the two logarithms into one.\newlineThe quotient rule states that log⁡b(a)−log⁡b(c)=log⁡b(ac)\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right). We can use this rule to combine our two logarithms into a single logarithm.\newlinelog⁡c((w−7)7)−log⁡c((w+2)3)\log_c\left((w-7)^7\right) - \log_c\left((w+2)^3\right) becomes log⁡c((w−7)7(w+2)3)\log_c\left(\frac{(w-7)^7}{(w+2)^3}\right)

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