Logarithmic Equations: Problems with Solutions

By Denitsa Dimitrova (Bulgaria)
Problem 1
$\log_3(3x-2)=1$
Problem 2
$\log_2(3x^2-2x-1)=5$
Problem 3
$\log_5(x^2-2x-2)=0$
Problem 4
$\log_3\frac{x-2}{x+3}=1$
Problem 5
$\log_{0.5}(4x-x^2)=-2$
Problem 6
$\lg(3x^2-2x) = \lg(5x-4)$
Problem 7
$\log_2(x+2)-\log_2x=1$
Problem 8
$\log x+ \log(3-x)=\log8+\log(2x+5)$
Problem 9
$\lg(4x+10)-\lg(x+3)=2\lg4-\lg8$
Problem 10
$\log_{2x+3}\left(\frac14\right)+2=0$

Problem 11
$\frac{12}{\lg x}-1=\lg x$
Problem 12
$\frac{8}{3+\log_{10}x}+\frac{7}{1-\log_{10}x}=3$
Problem 13
$(2-\log_6x)\cdot\log_6x=0.75$
Problem 14
$\log_3x+\log_x9=3$
Problem 15
$\log_{10}^2x+\log_{10}x^2=3$
Problem 16
$\log_3(3^{2x}-3^x-63)=x$
Problem 17
$\log_{10}\sqrt{x+5}+\log_{10}\sqrt{3x-5}=1$
Problem 18
$\log_2(x-2)-2=6\log_{\frac{1}{8}}\sqrt{3x-5}$
Problem 19
$\log_2(\log_3(x^2-16))-\log_{\frac{1}{2}}(\log_{\frac{1}{3}}\frac{1}{x^2-16})=2$
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