Math  /  Algebra

QuestionWrite the expression as a single logarithm. 13logcx+3logcylogcz\frac{1}{3} \log _{c} x+3 \log _{c} y-\log _{c} z

Studdy Solution

STEP 1

1. We are given a logarithmic expression that needs to be combined into a single logarithm.
2. The properties of logarithms, such as the power rule, product rule, and quotient rule, can be used to simplify the expression.

STEP 2

1. Apply the power rule to each logarithmic term.
2. Combine the logarithmic terms using the product and quotient rules.

STEP 3

Apply the power rule to each term in the expression. The power rule states that alogbM=logbMa a \log_b M = \log_b M^a .
For the first term: 13logcx=logcx1/3 \frac{1}{3} \log_{c} x = \log_{c} x^{1/3}
For the second term: 3logcy=logcy3 3 \log_{c} y = \log_{c} y^3
The third term remains: logcz=logcz1 -\log_{c} z = \log_{c} z^{-1}

STEP 4

Combine the logarithmic terms using the product and quotient rules. The product rule states that logbM+logbN=logb(MN) \log_b M + \log_b N = \log_b (M \cdot N) , and the quotient rule states that logbMlogbN=logb(MN) \log_b M - \log_b N = \log_b \left(\frac{M}{N}\right) .
Combine the first two terms using the product rule: logcx1/3+logcy3=logc(x1/3y3) \log_{c} x^{1/3} + \log_{c} y^3 = \log_{c} (x^{1/3} \cdot y^3)
Now, combine with the third term using the quotient rule: logc(x1/3y3)logcz=logc(x1/3y3z) \log_{c} (x^{1/3} \cdot y^3) - \log_{c} z = \log_{c} \left(\frac{x^{1/3} \cdot y^3}{z}\right)
The expression as a single logarithm is: logc(x1/3y3z) \log_{c} \left(\frac{x^{1/3} \cdot y^3}{z}\right)

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