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قوانين اللوغاريتمات

(a) log_b 14 = log_b (2 x 7)

    = log_b 2 + log_b 7

    ≈ 0.43 + 1.21 ≈ 1.64

(b) log_b $\frac{2}{7}$ = log_b 2 - log_b 7

                  ≈ 0.43 - 1.21 ≈ -0.78

(c) log_b 32 = log_b 2⁵ = 5 log_b 2

                  ≈ 5 x 0.43 ≈ 2.15

(d) log_b $\frac{1}{49}$ = log_b 1 - log_b 49

                   = 0 - log_b 7²

                   ≈ 0 – 2 log_b 7

                   ≈ 0 - 2 x 1.21 ≈ -2.42

(a) log₂ a² b⁹ = log₂ a² + log₂ b⁹

                       = 2 log₂ a + 9 log₂ b

(b) log₅ $\frac{{(x + 1)}^3}{8}$ = log₅ (x + 1)³ – log₅ 8

                            = 3 log₅ (x + 1) – log₅ 8

(c) log₃ $\frac{x^7 y^3}{z^5}$ = log₃ x⁷ y³ – log₃ z⁵

                        = log₃ x⁷ + log₃ y³ – log₃ z⁵

                        = 7 log₃ x + 3 log₃ y – 5 log₃ z

(d) log_b $\sqrt[3]{\frac{x^7 b^2}{y^5}}$ = log_b ($\frac{x^7 b^2}{y^5}$)^{$\frac{1}{3}$}

                            = $\frac{1}{3}$ log_b $\frac{x^7 b^2}{y^5}$

                            = $\frac{1}{3}$ (log_b x⁷ b² - log_b y⁵)

                            = $\frac{1}{3}$ (log_b x⁷ + log_b b² – log_b y⁵)

                            =  $\frac{1}{3}$ (7 log_b x + 2 log_b b – 5 log_b y

                            =  $\frac{7}{3}$ log_b x + $\frac{2}{3}$log_b b – $\frac{5}{3}$log_b y

                            = $\frac{7}{3}$ log_b x + $\frac{2}{3}$ - $\frac{5}{3}$ log_b y

(a) log₅ a + 3 log₅ b = log₅ a + log₅ b³

                                 = log₅ ab³

(b) 5 log_b x + $\frac{1}{2}$ log_b y – 9 log_b z  

                       = log_b x⁵ + log_b $y^{\frac{1}{2}}$ - log_b z⁹

                       = log_b x⁵ $y^{\frac{1}{2}}$ - log_b  z⁹

                       = log_{b $\frac{x^5 y^{{\displaystyle \frac{1}{2}}} }{z^9}$}

                       = log_{b $\frac{x^5 \sqrt{y} }{z^9}$}

M(t) = 92 – 28 log₁₀ (t + 1)

M(29) = 92 – 28 log₁₀ (29 + 1)

            = 92 – 28 log₁₀ 30

            = 92 – 28 log₁₀ (10 x 3)

            = 92 – 28 (log₁₀ 10 + log₁₀ 3)

            ≈ 92 – 28 (1 + 0.4771)

            ≈ 92 – 28 (1.4771)

            ≈ 92 – 41.3588

            ≈ 51

النسبة المئوية للموضوعات التي يتذكرها هذا الطالب بعد 29 شهراً هي 51% تقريباً.