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Math  /  Algebra

QuestionExpress as a single logarithm. log⁡109−log⁡103log⁡109−log⁡103=\begin{array}{c} \log _{10} 9-\log _{10} 3 \\ \log _{10} 9-\log _{10} 3= \end{array}

Studdy Solution

STEP 1

1. We are dealing with logarithms of the same base, which is base 10.
2. We can use logarithmic properties to simplify the expression.

STEP 2

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

STEP 3

Apply the logarithmic property that states:
log⁡ba−log⁡bc=log⁡b(ac) \log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)
For the given expression:
log⁡109−log⁡103=log⁡10(93) \log_{10} 9 - \log_{10} 3 = \log_{10} \left(\frac{9}{3}\right)

STEP 4

Simplify the fraction inside the logarithm:
93=3 \frac{9}{3} = 3
Thus, the expression becomes:
log⁡103 \log_{10} 3
The expression as a single logarithm is:
log⁡103 \boxed{\log_{10} 3}

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