Solve

\(\begin{array}{rcll} log_{10}(x+3)-log_{10}x &=& 2 \\ log_{10}( \frac{x+3}{x} ) &=& 2 \\ \frac{x+3}{x} &=& 10^2 \\ \frac{x+3}{x} &=&100 \\ x+3 &=& 100x \qquad | \qquad -x \\ 3 &=& 99x \\ 99x &=& 3 \qquad | \qquad :99 \\ x &=& \frac{3}{99} \\ x &=& \frac{1}{33} \\ x &=& 0.03030303030 \end{array}\)

\(\begin{array}{rcll} log_{15}(5-n) &=& log_{15}(4n-5) \\ 5-n &=& 4n-5 \\ 5 &=& 5n-5 \\ 10 &=& 5n \\ n &=& \frac{10}{5} \\ n &=& 2 \end{array}\)

\(\begin{array}{rcll} log_{12}(2-3m) &=& log_{12}(-5m+2) \\ 2-3m &=& -5m+2 \\ -3m &=& -5m \\ 5m-3m &=& 0 \\ 2m &=& 0 \qquad | \qquad : 2\\ m &=& 0 \\ \end{array}\)

\(\begin{array}{rcll} log_2(5)-log_2(4x) &=& 3 \\ log_2( \frac{5}{4x} ) &=& 3 \\ \frac{5}{4x} &=& 2^3 \\ \frac{5}{4x} &=& 8 \qquad | \qquad \cdot 4x \\ 5 &=& 8 \cdot 4x \\ 5 &=& 32x \\ 32x &=& 5 \qquad | \qquad :32 \\ x &=& \frac{5}{32} \\ x &=& 0.15625 \end{array}\)

\(\begin{array}{rcll} log_3(x^2)-log_3(4) &=& 5 \\ log_3( \frac{x^2} {4} ) &=& 5 \\ \frac{x^2} {4} &=& 3^5 \\ \frac{x^2} {4} &=& 234 \\ x^2 &=& 4\cdot 234 \\ x^2 &=& 972 \\ x &=& 31.1769145362 \end{array}\)

\(\begin{array}{rcll} log_6(5x^2)-log_6(5) &=& 4 \\ log_6( \frac{5x^2} {5} ) &=& 4 \\ log_6( x^2 ) &=& 4 \\ x^2 &=& 6^4 \\ x &=& 6^2 \\ x &=& 36 \\ \end{array}\)

#4:

solution:

\(x=\pm\sqrt{972} \\ = \pm 18\sqrt{3}\\ =\pm 31.1769145362\)

#5:

solution:

\(x = \pm 36\)