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Step 1:: The first law of logarithms, also known as the product rule, states that the logarithm of a product is equal to the sum of the logarithms of the factors.
\log\_b(xy) = \log\_b(x) + \log\_b(y)
Step 2:: The second law, or quotient rule, states that the logarithm of a quotient is equal to the difference between the logarithms of the dividend and the divisor.
\log\_b\left(\frac{x}{y}\right) = \log\_b(x) - \log\_b(y)
Step 3:: The third law, or power rule, states that the logarithm of a number raised to an exponent is equal to the product of the exponent and the logarithm of the number.
\log\_b(x^y) = y\log\_b(x)
Step 4:: The fourth law, or change of base formula, states that any logarithm can be converted to a different base using the following formula:
\log\_a(x) = \frac{\log\_b(x)}{\log\_b(a)}
Step 5:: The fifth law states that the logarithm of 1 is always 0 for any base.
\log\_b(1) = 0
Step 6:: The sixth law states that the logarithm of the base is always 1.
\log\_b(b) = 1
Step 7:: The seventh law states that if the logarithm of a number is 0, then the number must be equal to 1.
\log\_b(x) = 0 \Rightarrow x = 1
Final Answer
The seven laws of logarithms are: 1. Product rule: \log\_b(xy) = \log\_b(x) + \log\_b(y) 2. Quotient rule: \log\_b\left(\frac{x}{y}\right) = \log\_b(x) - \log\_b(y) 3. Power rule: \log\_b(x^y) = y\log\_b(x) 4. Change of base formula: \log\_a(x) = \frac{\log\_b(x)}{\log\_b(a)} 5. Logarithm of 1: \log\_b(1) = 0 6. Logarithm of the base: \log\_b(b) = 1 7. If the logarithm of a number is 0, the number is 1: \log\_b(x) = 0 \Rightarrow x = 1
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