Hard Logarithm Problems With Solutions Pdf Here

Let (a = \ln x). Then (\ln(2x) = a + \ln 2), (\ln(4x) = a + 2\ln 2).

(0 < \log_2 A < 1 \Rightarrow 1 < A < 2 \Rightarrow 1 < x^2-5x+7 < 2).

Challenging Exercises for Advanced High School & Early College Students hard logarithm problems with solutions pdf

Title: Hard Logarithm Problems with Detailed Solutions

Change base: (\log_{x}(2x+3) = \frac{\ln(2x+3)}{\ln x}), (\log_{x+1}(x+2) = \frac{\ln(x+2)}{\ln(x+1)}). Let (a = \ln x)

Equation: (\ln 2 \cdot (a + 2\ln 2) = a \cdot (a + \ln 2)).

Cancel (a\ln 2) both sides: (2(\ln 2)^2 = a^2 \Rightarrow a = \pm \sqrt{2} \ln 2). Challenging Exercises for Advanced High School & Early

Answer: No real solution. Domain: (x>0, x\neq 1, 2x>0, 2x\neq 1, 4x>0, 4x\neq 1) → (x>0, x\neq 1, x\neq 0.5, x\neq 0.25).

Use (\log A + \log B = \log(AB)): [ \log_5 \left[ (x^2 - 4x + 5)(x^2 + 4x + 5) \right] = 2 ] But ((a-b)(a+b) = a^2 - b^2): Let (a=x^2+5), (b=4x): [ (x^2+5 - 4x)(x^2+5+4x) = (x^2+5)^2 - (4x)^2 = x^4 + 10x^2 + 25 - 16x^2 ] [ = x^4 - 6x^2 + 25 ] So: [ \log_5 (x^4 - 6x^2 + 25) = 2 ] [ x^4 - 6x^2 + 25 = 5^2 = 25 ] [ x^4 - 6x^2 = 0 \quad \Rightarrow \quad x^2(x^2 - 6) = 0 ] (x=0) or (x=\pm\sqrt{6}).

Convert to natural logs: (\log_x 2 = \frac{\ln 2}{\ln x}), (\log_{2x} 2 = \frac{\ln 2}{\ln(2x)}), (\log_{4x} 2 = \frac{\ln 2}{\ln(4x)}).

Convert to base 10 (or natural log): Let (\ln x = t). (\log_2 x = \frac{t}{\ln 2}), (\log_3 x = \frac{t}{\ln 3}), (\log_4 x = \frac{t}{\ln 4} = \frac{t}{2\ln 2}).