log3(x+17)−log3(x−9)=3 rewrite the equation without logarithms. Do not solve for x

To rewrite the equation without logarithms, we will apply the properties of logarithms, specifically the **logarithmic difference rule**:

\[
\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)
\]

Given the equation:

\[
\log_3(x + 17) - \log_3(x - 9) = 3
\]

Using the difference rule:

\[
\log_3\left(\frac{x + 17}{x - 9}\right) = 3
\]

Now, to eliminate the logarithm, we rewrite the equation in exponential form. Recall that:

\[
\log_b(A) = C \quad \text{is equivalent to} \quad A = b^C
\]

So,

\[
\frac{x + 17}{x - 9} = 3^3
\]

Since \( 3^3 = 27 \), the equation becomes:

\[
\frac{x + 17}{x - 9} = 27
\]

This is the equation rewritten without logarithms.

Would you like further clarification on this? Here are five related questions to deepen your understanding:

1. What are the key logarithmic properties used to rewrite the equation?
2. How can the equation \(\frac{x + 17}{x - 9} = 27\) be solved for \(x\)?
3. What is the general rule for converting logarithmic equations to exponential form?
4. How does the base of the logarithm affect the rewriting process?
5. How could this process change if the equation involved a natural logarithm (log base \(e\))?

**Tip:** Always check if the arguments of a logarithmic function are valid. In this case, both \(x + 17\) and \(x - 9\) must be positive.

Learn how to rewrite the logarithmic equation log3(x+17) − log3(x−9) = 3 without using logarithms. This problem explores key concepts in logarithms, exponential equations, and algebra, and is suitable for students in Grades 9-12.

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