Log Problems for solving logarithmic equations

You have learned the logarithmic equations and it is your time to learn how to solve any logarithmic equation easily from the list of below solved logarithm problems. If you are a beginner, it’s very useful in solving logarithmic equations. It is also useful for practicing if you are an advanced learner.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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