Determine the value of the unknown.
3 log Subscript 125 Baseline left parenthesis Upper A minus 6 right parenthesis equals negative 1
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Aequals
  
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​(Simplify your​ answer.)

We are given the equation:

\[
3 \log_{125} (A - 6) = -1
\]

To solve for \( A \), let's go through the following steps:

### Step 1: Isolate the logarithm
First, divide both sides by 3 to simplify the equation:

\[
\log_{125} (A - 6) = \frac{-1}{3}
\]

### Step 2: Convert from logarithmic to exponential form
The logarithmic equation \(\log_b(x) = y\) can be rewritten in exponential form as \(x = b^y\). Using this property, we rewrite the equation:

\[
A - 6 = 125^{\frac{-1}{3}}
\]

### Step 3: Simplify the exponential expression
\(125^{\frac{1}{3}}\) is the cube root of 125, which equals 5. So:

\[
125^{\frac{-1}{3}} = \frac{1}{5}
\]

Now the equation becomes:

\[
A - 6 = \frac{1}{5}
\]

### Step 4: Solve for \(A\)
Add 6 to both sides:

\[
A = 6 + \frac{1}{5}
\]

Convert 6 to a fraction:

\[
A = \frac{30}{5} + \frac{1}{5} = \frac{31}{5}
\]

### Final Answer:
\[
A = \frac{31}{5}
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you convert a logarithmic equation to an exponential form?
2. What is the value of \( \log_b(x) \) when \( x \) is a fraction?
3. How do you simplify fractional exponents?
4. Can logarithms have negative values?
5. How do you add fractions with different denominators?

**Tip:** When solving logarithmic equations, always check if the argument of the log stays positive in the final solution.

Determine the value of the unknown. 3 log Subscript 125 Baseline left parenthesis Upper A minus 6 right parenthesis equals negative 1. (Simplify your answer.)

Learn how to solve the logarithmic equation 3 log_125(A - 6) = -1 with a clear step-by-step explanation. This solution covers key logarithmic concepts, including converting logarithmic expressions to exponential form and solving for the unknown. Suitable for students in Grades 9-12.

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