Question

solve for x.
\log_{5} 8 = \log_{5} 2 + \log_{5} (x + 6)

Explanation:

Step1: Apply log addition rule

Using the property \(\log_a b+\log_a c = \log_a(bc)\), rewrite the right - hand side:
\(\log_{5}8=\log_{5}[2(x + 6)]\)

Step2: Use one - to - one property of logs

If \(\log_a m=\log_a n\), then \(m = n\) (for \(a>0,a
eq1,m>0,n>0\)). So we have:
\(8=2(x + 6)\)

Step3: Solve the linear equation

First, divide both sides by 2:
\(\frac{8}{2}=\frac{2(x + 6)}{2}\)
\(4=x + 6\)
Then, subtract 6 from both sides:
\(x=4 - 6=- 2\)

Step4: Check the domain

For \(\log_{5}(x + 6)\), we need \(x+6>0\). When \(x=-2\), \(x + 6=-2 + 6 = 4>0\), and \(\log_{5}2\) and \(\log_{5}8\) are well - defined.

Answer:

\(x=-2\)