12 Patterns of Logarithmic Equations

Solving logarithmic equations is done many ways using properties of logarithmic functions, such as multiply of logs, change the base and reciprocals of logarithms.
$$ \begin{aligned} \displaystyle \large a^x = y \ &\large \Leftrightarrow x = \log_{a}{y} \\
\large \log{a} + \log{b} &= \large \log{(a \times b)} \\
\large \log{a} – \log{b} &= \large \log{(a \div b)} \\
\large \log{a^n} &= \large n \log{a} \\
\large \log_{a}{b} &= \large \frac{\log_{c}{a}}{\log_{c}{b}} \\
\large \log_{a}{b} &= \large \frac{1}{\log_{b}{a}} \\
\large \log_{a}{a} &= \large 1 \\
\end{aligned} \\ $$

Worked Examples of Logarithmic Equations

Question 1

Solve \( 5 – \log_{4}{8} = \log_{4}{x} \).

Question 2

Solve \( \log_{10}{x} + \log_{10}{(x-3)} = \log_{10}{4} \).

Question 3

Solve \( \log_{4}{x} + \log_{4}{(x-6)} = 2 \).

Question 4

Solve \( \big(\log_{10}{x}\big)^2 – 2 \log_{10}{x} – 3 = 0 \).

Question 5

Solve \( \big(\log_{3}{x}\big)^2 – \log_{3}{x^4} + 3 = 0 \).

Question 6

Solve \( \big(\log_{2}{x}\big)^2 = \log_{2}{x^4} \).

Question 7

Solve \( \displaystyle \frac{\log_{10}{x}}{\log_{10}{2}} = 4 \).

Question 8

Solve \( \log_{10}{x^2} + \log_{10}{8x} = 3. \)

Question 9

Solve \( \log_{4}{x} – \log_{8}{x} = 2 \).

Question 10

Solve \( \log_{2}{x} – \log_{x}{4} = 1 \).

Question 11

Solve \( x^{\log_{10}{x}} = 1000x^2 \).

Question 12

Solve \( 5^{\log_{10}{x}} \times x^{\log_{10}{5}} – 3 (5^{\log_{10}{x}} + x^{\log_{10}{5}}) + 5 = 0 \).