\(\log_b{(xy)}=\log_b{(x)}+\log_b{(y)}\)

\(\log_b{\left( \dfrac{x}{y} \right)}=\log_b{(x)}-\log_b{(y)}\)

\(\log_b{(x^n)}=n\log_b{(x)}\)

\(\log_b{(x})=\dfrac{\log_k{(x)}}{\log_k{(b)}}\) (for any base \(k\))

• Common Logarithm: Base \(10\), denoted as \(\log{(x)}\).
• Natural Logarithm: Base \(e\) (Euler's number, approximately \(2.718\)), denoted as \(\ln{(x)}\).

• Converting to Exponential Form: Use the definition \(\log_b{x}=y \Rightarrow b^y=x\).
• Applying Logarithm Laws: Simplify and solve equations using the product, quotient, and power laws.
• Example:
  \(\begin{aligned} \log_2{(x)}+\log_2{(x-1)}&=3 \\ \log_2{(x(x-1))}&=3 \\ x(x-1)&=8 \end{aligned}\)

\(\log_b{(1)}=0\) (since \(b^0=1\))

\(\log_b{(b)}=1\) (since \(b^1=b\))

• Shape: The graph of \(y=\log_a(x)\) is a curve that increases slowly and passes through the point \((1,0)\).
• Asymptote: The vertical line \(x=0\) is an asymptote.
• Domain and Range: The domain are \(x>0\), while the range are all real numbers.
• Relation with Exponential Function:  The exponential and logarithmic functions are reflection of one another in the straight line \(y=x\). The exponential and logarithmic functions are inverse functions of one another.

Generally, 

If \(f:x\rightarrow a^x\), then \(f^{-1}:x\rightarrow \log_ax\).

Thus,

\(y=\log_ax\) is the inverse of \(a^y=x\).

Solve the equation \(3^{x-4}=50^{x-3}\).

\(\begin{aligned} 3^{x-4}&=50^{x-3} \\ (x-4)\log3&=(x-3)\log50 \\ x\log3-4\log3&=x\log50-3\log50 \\ x\log3-x\log50&=-3\log50+4\log3 \\ x(\log3-\log50)&=-3\log50+4\log3 \\ x&=\dfrac{-3\log50+4\log3}{\log3-\log50} \\ &=2.610 .\end{aligned}\)

Thus, \(x=2.610\) is the solution for the equation.

\(\log_b{(xy)}=\log_b{(x)}+\log_b{(y)}\)

\(\log_b{\left( \dfrac{x}{y} \right)}=\log_b{(x)}-\log_b{(y)}\)

\(\log_b{(x^n)}=n\log_b{(x)}\)

\(\log_b{(x})=\dfrac{\log_k{(x)}}{\log_k{(b)}}\) (for any base \(k\))

• Common Logarithm: Base \(10\), denoted as \(\log{(x)}\).
• Natural Logarithm: Base \(e\) (Euler's number, approximately \(2.718\)), denoted as \(\ln{(x)}\).

• Converting to Exponential Form: Use the definition \(\log_b{x}=y \Rightarrow b^y=x\).
• Applying Logarithm Laws: Simplify and solve equations using the product, quotient, and power laws.
• Example:
  \(\begin{aligned} \log_2{(x)}+\log_2{(x-1)}&=3 \\ \log_2{(x(x-1))}&=3 \\ x(x-1)&=8 \end{aligned}\)

\(\log_b{(1)}=0\) (since \(b^0=1\))

\(\log_b{(b)}=1\) (since \(b^1=b\))

• Shape: The graph of \(y=\log_a(x)\) is a curve that increases slowly and passes through the point \((1,0)\).
• Asymptote: The vertical line \(x=0\) is an asymptote.
• Domain and Range: The domain are \(x>0\), while the range are all real numbers.
• Relation with Exponential Function:  The exponential and logarithmic functions are reflection of one another in the straight line \(y=x\). The exponential and logarithmic functions are inverse functions of one another.

Generally, 

If \(f:x\rightarrow a^x\), then \(f^{-1}:x\rightarrow \log_ax\).

Thus,

\(y=\log_ax\) is the inverse of \(a^y=x\).

Solve the equation \(3^{x-4}=50^{x-3}\).

\(\begin{aligned} 3^{x-4}&=50^{x-3} \\ (x-4)\log3&=(x-3)\log50 \\ x\log3-4\log3&=x\log50-3\log50 \\ x\log3-x\log50&=-3\log50+4\log3 \\ x(\log3-\log50)&=-3\log50+4\log3 \\ x&=\dfrac{-3\log50+4\log3}{\log3-\log50} \\ &=2.610 .\end{aligned}\)

Thus, \(x=2.610\) is the solution for the equation.