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Math  /  Algebra

QuestionExpress as a single logarithmic expression. You do NOT need to expand exponents. Assume all expressions represent positive numbers. log⁡6(x+1)−log⁡6(x+5)=log⁡6(□)\log _{6}(x+1)-\log _{6}(x+5)=\log _{6}(\square)
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Studdy Solution

STEP 1

1. We are working with logarithms of the same base, which is 6.
2. The properties of logarithms can be applied to combine the expression into a single logarithm.
3. All expressions represent positive numbers, so the logarithms are defined.

STEP 2

1. Apply the logarithmic property for subtraction.
2. Simplify the expression.

STEP 3

Apply the property of logarithms that states:
log⁡b(A)−log⁡b(B)=log⁡b(AB) \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)
In this case, apply it to the expression:
log⁡6(x+1)−log⁡6(x+5)=log⁡6(x+1x+5) \log_{6}(x+1) - \log_{6}(x+5) = \log_{6}\left(\frac{x+1}{x+5}\right)

STEP 4

The expression is already simplified as a single logarithmic expression:
log⁡6(x+1x+5) \log_{6}\left(\frac{x+1}{x+5}\right)
This is the final expression for the given problem.
The single logarithmic expression is:
log⁡6(□)=log⁡6(x+1x+5) \log_{6}(\square) = \log_{6}\left(\frac{x+1}{x+5}\right)

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