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 logarithm addition rule

The logarithm addition rule states that \(\log_b M + \log_b N=\log_b(MN)\) for the same base \(b\). We can apply this rule to combine the three logarithms.
First, let's consider the exponents of \(e\) in each argument. The first term is \(\log_y e^6\), the second is \(\log_y e=\log_y e^1\), and the third is \(\log_y e^5\).

Step2: Combine the logarithms using the rule

Using the property \(\log_b a^n = n\log_b a\) (power rule) and the addition rule. But since we are adding logarithms with the same base, we can multiply their arguments.
The arguments are \(e^6\), \(e^1\), and \(e^5\). When we multiply \(e^6\times e^1\times e^5\), we use the exponent rule \(a^m\times a^n=a^{m + n}\). So \(6+1 + 5=12\), so \(e^6\times e\times e^5=e^{6 + 1+5}=e^{12}\).
Then, by the addition rule of logarithms \(\log_y e^6+\log_y e+\log_y e^5=\log_y(e^6\times e\times e^5)=\log_y e^{12}\)

Answer:

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