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Solve the equation. Give the solution in exact form.

[tex]\[ \log (x-8)-\log (x-2)=\log 3 \][/tex]

Rewrite the given equation without logarithms. Do not solve for \( x \).

[tex]\[ \square \][/tex]

Sagot :

GinnyAnswer
Certainly! Let's solve the given logarithmic equation step-by-step and rewrite it without logarithms.

The initial equation is:
[tex]\[ \log (x-8) - \log (x-2) = \log 3 \][/tex]

1. Apply the properties of logarithms: Recall that \(\log a - \log b = \log \left( \frac{a}{b} \right) \). Use this property to combine the logarithms on the left side.
[tex]\[ \log \left( \frac{x-8}{x-2} \right) = \log 3 \][/tex]

2. Exponentiate both sides to remove the logarithms: Since the logarithms are equal, their arguments must also be equal.
[tex]\[ \frac{x-8}{x-2} = 3 \][/tex]

Therefore, the equation rewritten without logarithms is:
[tex]\[ \frac{x-8}{x-2} = 3 \][/tex]
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