# 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} \\

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