Solve the following equations.

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Solve the following equations. (i) x1+log⁡10x=10xx^{1+\log_{10}x}=10x (ii) log⁡2(9+2x)=3\log_2(9+2^x)=3 (iii) 2xlog⁡43+3log⁡4x=272x^{\log_4 3}+3^{\log_4 x}=27 (iv) log⁡4log⁡3log⁡2x=0\log_4\log_3\log_2 x=0 (v) log⁡10x+5x3=105+log⁡10x\frac{\log_{10}x+5}{x^3}=10^{5+\log_{10}x} (vi) log⁡3(log⁡9x+12+9x)=2x\log_3\left(\log_9x+\frac{1}{2}+9^x\right)=2x (vii) 4log⁡10x+1−6log⁡10x−2⋅3log⁡10x2+2=04^{\log_{10}x+1}-6^{\log_{10}x}-2\cdot3^{\log_{10}x^2+2}=0 (viii) log⁡10(x−3)log⁡10(x2−21)=12\frac{\log_{10}(x-3)}{\log_{10}(x^2-21)}=\frac{1}{2} (ix) xlog⁡2x+4=32x^{\log_2x+4}=32 (x) log⁡ax=x\log_a x=x, where a=xlog⁡4xa=x^{\log_4x} (xi) log⁡2(sin⁡x(1+cos⁡x))=2\log_{\sqrt{2}}\left(\sin x(1+\cos x)\right)=2

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