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Original
2026-01-01
Modified
2026-02-28
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<p>Each property of<a>logarithms</a>aims to simplify and solve logarithmic equations and<a>expressions</a>.</p>
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<p>Each property of<a>logarithms</a>aims to simplify and solve logarithmic equations and<a>expressions</a>.</p>
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<p>For all logarithmic properties: (m, n > 0, a > 0, a ≠ 1) Only positive<a>real numbers</a>can be used to define logarithms, and their bases must be positive and<a>not equal</a>to 1. While more properties exist, the four basic properties are listed below: </p>
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<p>For all logarithmic properties: (m, n > 0, a > 0, a ≠ 1) Only positive<a>real numbers</a>can be used to define logarithms, and their bases must be positive and<a>not equal</a>to 1. While more properties exist, the four basic properties are listed below: </p>
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<p><strong>Product property: </strong></p>
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<p><strong>Product property: </strong></p>
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<p>\( log_a mn = log_a m + log_a n \) </p>
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<p>\( log_a mn = log_a m + log_a n \) </p>
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<p>Condition: \((m, n > 0, a > 0, a ≠ 1)\) </p>
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<p>Condition: \((m, n > 0, a > 0, a ≠ 1)\) </p>
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<p>This property states that the logarithm of a<a>product</a>\((mn)\) is equal to the<a>sum</a>of the logarithms of the individual<a>factors</a>. </p>
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<p>This property states that the logarithm of a<a>product</a>\((mn)\) is equal to the<a>sum</a>of the logarithms of the individual<a>factors</a>. </p>
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<p><strong>Quotient property:</strong></p>
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<p><strong>Quotient property:</strong></p>
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<p> \(log_a m / n = log_a m - log_a n \) </p>
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<p> \(log_a m / n = log_a m - log_a n \) </p>
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<p>Condition: \((m, n > 0, a > 0, a ≠ 1)\) </p>
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<p>Condition: \((m, n > 0, a > 0, a ≠ 1)\) </p>
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<p>The quotient property states that the logarithm of a quotient (m/n) is the same as the difference between the logarithms of numerator and denominator.</p>
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<p>The quotient property states that the logarithm of a quotient (m/n) is the same as the difference between the logarithms of numerator and denominator.</p>
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<p><strong>Power property:</strong></p>
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<p><strong>Power property:</strong></p>
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<p>\(log_a(m^n) = n log_a(m)\) </p>
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<p>\(log_a(m^n) = n log_a(m)\) </p>
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<p>Condition: \((m, n > 0, a > 0, a ≠ 1)\) </p>
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<p>Condition: \((m, n > 0, a > 0, a ≠ 1)\) </p>
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<p>The rule shows that if we move the exponent inside a logarithm to the front of a log, then the exponent will be the multiplier. </p>
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<p>The rule shows that if we move the exponent inside a logarithm to the front of a log, then the exponent will be the multiplier. </p>
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<p><strong>Change of base property:</strong></p>
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<p><strong>Change of base property:</strong></p>
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<p> \(log_b a = log_c a / log_c b \) </p>
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<p> \(log_b a = log_c a / log_c b \) </p>
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<p>Condition: \((m, n > 0, a > 0, a ≠ 1)\) </p>
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<p>Condition: \((m, n > 0, a > 0, a ≠ 1)\) </p>
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<p>This property states that we can convert the base of a logarithm to another base. </p>
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<p>This property states that we can convert the base of a logarithm to another base. </p>
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<p>Other properties are also directly derived from exponent rules. The definition of logarithm is: </p>
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<p>Other properties are also directly derived from exponent rules. The definition of logarithm is: </p>
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<p>\(a^x = m\) ⇔ \(log_a m = x\)</p>
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<p>\(a^x = m\) ⇔ \(log_a m = x\)</p>
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<p>\(a^0 = 1\) ⇒ \(log_a 1 = 0\) </p>
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<p>\(a^0 = 1\) ⇒ \(log_a 1 = 0\) </p>
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<p>\(a^1 = a\) ⇒ \(log_a a = 1\) </p>
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<p>\(a^1 = a\) ⇒ \(log_a a = 1\) </p>
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<p>\(a^{\log_a x} = x \) </p>
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<p>\(a^{\log_a x} = x \) </p>
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<p>\(\log_b (a^m) = m \cdot \log_b (a) \)</p>
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<p>\(\log_b (a^m) = m \cdot \log_b (a) \)</p>
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<p>(Results from changing the power property and base rule). </p>
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<p>(Results from changing the power property and base rule). </p>
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<p>The following table lists the properties of logarithms. </p>
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<p>The following table lists the properties of logarithms. </p>
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