Question

rewrite the logarithmic expression as a single logarithm with the same base. assume all expressions exist and are well - defined. \\(\log_{y}e^{6}+\log_{y}e + \log_{y}e^{5}\\)

Explanation:

Step1: Recall the logarithmic addition rule

The rule for adding logarithms with the same base is \(\log_b M + \log_b N=\log_b(MN)\). We can apply this rule to the given expression \(\log_{y}e^{6}+\log_{y}e+\log_{y}e^{5}\). First, we can combine the first two logarithms: \(\log_{y}e^{6}+\log_{y}e=\log_{y}(e^{6}\times e)\).

Step2: Simplify the product inside the logarithm

Using the exponent rule \(a^{m}\times a^{n}=a^{m + n}\), we have \(e^{6}\times e=e^{6 + 1}=e^{7}\). So now our expression becomes \(\log_{y}e^{7}+\log_{y}e^{5}\).

Step3: Combine the remaining two logarithms

Again, using the logarithmic addition rule \(\log_{y}e^{7}+\log_{y}e^{5}=\log_{y}(e^{7}\times e^{5})\).

Step4: Simplify the product of exponents

Using the exponent rule \(a^{m}\times a^{n}=a^{m + n}\) again, we get \(e^{7}\times e^{5}=e^{7+5}=e^{12}\). So the expression simplifies to \(\log_{y}e^{12}\).

Answer:

\(\log_{y}e^{12}\)