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Math  /  Algebra

QuestionWRITE EXPRESSION AS SINGLE LOGARITHM: A) 2log⁡2+3log⁡x−12[log⁡(x+3)+log⁡(x−2)]2 \log 2+3 \log x-\frac{1}{2}[\log (x+3)+\log (x-2)]

Studdy Solution

STEP 1

1. We are asked to combine multiple logarithmic terms into a single logarithm.
2. The properties of logarithms, such as the product, quotient, and power rules, can be used to simplify the expression.

STEP 2

1. Apply the power rule to each logarithmic term.
2. Use the product rule to combine terms with addition.
3. Use the quotient rule to combine terms with subtraction.
4. Simplify the expression to a single logarithm.

STEP 3

Apply the power rule to each logarithmic term. The power rule states that alog⁡b=log⁡ba a \log b = \log b^a .
2log⁡2=log⁡22=log⁡4 2 \log 2 = \log 2^2 = \log 4
3log⁡x=log⁡x3 3 \log x = \log x^3
12[log⁡(x+3)+log⁡(x−2)]=log⁡(x+3)1/2+log⁡(x−2)1/2 \frac{1}{2}[\log (x+3) + \log (x-2)] = \log (x+3)^{1/2} + \log (x-2)^{1/2}

STEP 4

Use the product rule to combine terms with addition. The product rule states that log⁡a+log⁡b=log⁡(a×b) \log a + \log b = \log (a \times b) .
log⁡(x+3)1/2+log⁡(x−2)1/2=log⁡((x+3)1/2×(x−2)1/2) \log (x+3)^{1/2} + \log (x-2)^{1/2} = \log \left((x+3)^{1/2} \times (x-2)^{1/2}\right)
=log⁡((x+3)(x−2)) = \log \left(\sqrt{(x+3)(x-2)}\right)

STEP 5

Use the quotient rule to combine terms with subtraction. The quotient rule states that log⁡a−log⁡b=log⁡(ab) \log a - \log b = \log \left(\frac{a}{b}\right) .
2log⁡2+3log⁡x−12[log⁡(x+3)+log⁡(x−2)] 2 \log 2 + 3 \log x - \frac{1}{2}[\log (x+3) + \log (x-2)]
=log⁡4+log⁡x3−log⁡((x+3)(x−2)) = \log 4 + \log x^3 - \log \left(\sqrt{(x+3)(x-2)}\right)
=log⁡(4x3(x+3)(x−2)) = \log \left(\frac{4x^3}{\sqrt{(x+3)(x-2)}}\right)

STEP 6

Simplify the expression to a single logarithm:
The expression as a single logarithm is:
log⁡(4x3(x+3)(x−2)) \log \left(\frac{4x^3}{\sqrt{(x+3)(x-2)}}\right)

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