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Math

QuestionIdentify which statements are correct:
I. (log⁡3x)13=13log⁡3x(\log_{3} x)^{\frac{1}{3}}=\frac{1}{3} \log_{3} x
II. log⁡3(pq5)=log⁡3p+5log⁡3q\log_{3}(p q^{5})=\log_{3} p+5 \log_{3} q
III. log⁡7alog⁡7b=log⁡7a−log⁡7b\frac{\log_{7} a}{\log_{7} b}=\log_{7} a-\log_{7} b

Studdy Solution

STEP 1

Assumptions1. The base of the logarithm is the same in all parts of each equation. . The properties of logarithms apply. These include - The power rule log⁡bmn=nlog⁡bm\log_b m^n = n \log_b m - The product rule log⁡bmn=log⁡bm+log⁡bn\log_b mn = \log_b m + \log_b n - The quotient rule log⁡bmn=log⁡bm−log⁡bn\log_b \frac{m}{n} = \log_b m - \log_b n

STEP 2

Let's examine the first equation(log⁡x)1=1log⁡x\left(\log{} x\right)^{\frac{1}{}}=\frac{1}{} \log{} xThe left side of the equation is raising the logarithm to the power of 1\frac{1}{}, which is not the same as multiplying the logarithm by 1\frac{1}{}, as shown on the right side. Therefore, the first equation is not correct.

STEP 3

Now, let's examine the second equationlog⁡3(pq5)=log⁡3p+5log⁡3q\log{3}\left(p q^{5}\right)=\log{3} p+5 \log{3} qThis equation is using the product rule of logarithms, which states that the logarithm of a product is equal to the sum of the logarithms of its factors. Therefore, the second equation is correct.

STEP 4

Finally, let's examine the third equationlog⁡7alog⁡7b=log⁡7a−log⁡7b\frac{\log{7} a}{\log{7} b}=\log{7} a-\log{7} bThis equation is trying to apply the quotient rule of logarithms, but it is not applied correctly. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Therefore, the third equation is not correct.
In conclusion, only the second equation is correct.

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