# Logarithmic equations worksheet

Test your skill on logarithms by solving following logarithmic equations and check your answer with our solution for improving your knowledge on solving logarithmic equations mathematically.

$(1) \,\,\,\,\,\,$ $\log_{3}{(5x-2)}$ $-$ $2\log_{3}{\sqrt{3x+1}}$ $=$ $1-\log_{3}{4}$

$(2) \,\,\,\,\,\,$ $\dfrac{\log_{2}{(9-2^x)}}{3-x}$ $\,=\,$ $1$

$(3) \,\,\,\,\,\,$ $2\log_{2+\sqrt{3}} {(\sqrt{x^2+1}+x)}$ $\,+\,$ $\log_{2-\sqrt{3}} {(\sqrt{x^2+1}-x)}$ $\,=\,$ $3$

$(4) \,\,\,\,\,\,$ $\log_{x} 2 \times \log_{\frac{x}{16}}{2}$ $\,=\,$ $\log_{\frac{x}{64}}{2}$

$(5) \,\,\,\,\,\,$ $2\log_{x}{a}$ $+$ $\log_{ax}{a}$ $+$ $3\log_{a^2x}{a}$ $\,=\,$ $0$

$(6) \,\,\,\,\,\,$ $x+\log{(1+2^x)}$ $\,=\,$ $x\log{5}$ $+$ $\log{6}$

$(7) \,\,\,\,\,\,$ $x^{(\log_{2}{x})+4} \,=\, 32$

$(8) \,\,\,\,\,\,$ $x^{(\log_{2}{x})+4} \,=\, 32$

$(9) \,\,\,\,\,\,$ $\log_{5-x}{(x^2 -2x+65)}$ $\,=\,$ $2$

$(10) \,\,\,\,\,\,$ $\log_{5}{x}$ $+$ $\log_{x}{5}$ $\,=\,$ $\dfrac{5}{2}$

$(11) \,\,\,\,\,\,$ $\dfrac{x}{y}$ $+$ $\dfrac{y}{x}$ If $\log \Bigg[\dfrac{x+y}{3}\Bigg]$ $\,=\,$ $\dfrac{1}{2} (\log x + \log y)$

$(12) \,\,\,\,\,\,$ $\dfrac{\log(\sqrt{x+1}+1)}{\log \sqrt[3]{x-40}}$ $\,=\,$ $3$

$(13) \,\,\,\,\,\,$ $\log_{10} \Big[98$ $+$ $\sqrt{x^2-12x+36}\Big]$ $\,=\,$ $2$

$(14) \,\,\,\,\,\,$ $\log_{2}{x}$ $+$ $\log_{4}{x}$ $+$ $\log_{16}{x}$ $\,=\,$ $\dfrac{21}{4}$

$(15) \,\,\,\,\,\,$ $\log{7}$ $+$ $\log{(3x-2)}$ $\,=\,$ $\log{(x+3)}+1$