# Logarithmic Equations Reducible to Quadratics

Logarithmic Equations Reducible to Quadratic of Math Online Tutoring is based on the basic properties of logarithms such as;
\large \displaystyle \begin{align} \log_{a}{b} &= \frac{1}{\log_{b}{a}} \\ \log_{a}{b} &= \frac{\log_{c}{b}}{\log_{c}{a}} \end{align}

## Question 1

Solve $2 \log_{2}{x}-9 \log_{x}{2} = 3$.

\begin{aligned} \displaystyle 2 \log_{2}{x}-\frac{9}{\log_{2}{x}} &= 3 \\ 2 (\log_{2}{x})^2-9 &= 3 \log_{2}{x} \\ 2 (\log_{2}{x})^2-3 \log_{2}{x}-9 &= 0 \\ (2 \log_{2}{x} + 3)(\log_{2}{x}-3) &= 0 \\ \log_{2}{x} = -\frac{3}{2} &\text{ or } \log_{2}{x} = 3 \\ x = 2^{-\frac{3}{2}} &\text{ or } x = 2^3 \\ \therefore x = \frac{1}{\sqrt{2^3}} &\text{ or } x = 8 \end{aligned}

## Question 2

Solve $\log{2x} + \log{(x-1)} = \log{(x^2+3)}$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} \log{2x(x-1)} &= \log{(x^2+3)} &\color{red} \log{A} + \log{B} = \log{AB} \\ 2x(x-1) &= x^2 + 3 \\ x^2-2x-3 &= 0 \\ (x-3)(x+1) &= 0 \\ x = 3 &\text{ or } x = -1 \\ \therefore x &= 3 &\color{red} x \gt 0 \text{ from } \log{2x} \end{aligned}

## Question 3

Solve $\log_{2}{(x-5)} = \log_{4}{(x-2)} + 1$.

\begin{aligned} \displaystyle \require{AMSsymbols} \log_{2}{(x-5)} &= \frac{\log_{2}{(x-2)}}{\log_{2}{4}} + 1 \\ \log_{2}{(x-5)} &= \frac{\log_{2}{(x-2)}}{2} + 1 \\ 2 \log_{2}{(x-5)} &= \log_{2}{(x-2)} + 2 \\ \log_{2}{(x-5)^2} &= \log_{2}{(x-2)} + \log_{2}{4} \\ \log_{2}{(x-5)^2} &= \log_{2}{4(x-2)} \\ (x-5)^2 &= 4(x-2) \\ x^2-10x + 25 &= 4x-8 \\ x^2-14x + 33 &= 0 \\ (x-3)(x-11) &= 0 \\ x = 3 \text{ or } x &= 11 \\ \therefore x &= 11 &\color{red} x \gt 5 \text{ from } \log_{2}{(x-5)} \end{aligned}

## Question 4

Solve $3 \log_{x}{10} + \log_{10}{x} = 4$.

\begin{aligned} \displaystyle \frac{3}{\log_{10}{x}} + \log_{10}{x} &= 4 \\ 3 + (\log_{10}{x})^2 &= 4 \log_{10}{x} \\ (\log_{10}{x})^2-4 \log_{10}{x} +3 \\ (\log_{10}{x}-1)(\log_{10}{x}-3) &= 0 \\ \log_{10}{x} = 1 &\text{ or } \log_{10}{x} = 3 \\ x = 10^1 &\text{ or } x = 10^3 \\ \therefore x = 10 &\text{ or } x = 1000 \end{aligned}

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