## Laws of Logarithms

 4.3 Laws of Logarithms
 Equation in the form of index and logarithm:

$$N=a^x \iff \log_aN=x$$

 where $$a \gt0,\, a \ne1$$. Logarithms:

\begin{aligned} &\bullet \log_aa^x=x \\\\ &\bullet \log_a1=0 \\\\ &\bullet \log_aa=1 \end{aligned}

 The diagram shows the graphs of exponential and logarithmic functions.

 We can see that 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. The logarithms of negative numbers and of zero are undefined. Law of logarithms:

\begin{aligned} &\bullet \log_axy=\log_ax+\log_ay \\\\ &\bullet \log_a\dfrac{x}{y}=\log_ax-\log_ay\\\\ &\bullet \log_ax^n=n \log_ax \end{aligned}

 for any real number $$n$$ where $$a,x$$ and $$y$$ are positive numbers and $$a\ne1$$. Change of base of logarithms:

\begin{aligned} &\bullet \log_ab=\dfrac{\log_cb}{\log_ca} \\\\ &\bullet \log_ab=\dfrac{1}{\log_ba} \end{aligned}

 where $$a$$, $$b$$ and $$c$$ are positive numbers, $$a \ne 1$$ and $$c \ne 1$$. $$\lg=\log_{10}$$ (common logarithms) and $$\ln=\log_{e}$$ (natural logarithms) where $$e$$ is a constant.

## Laws of Logarithms

 4.3 Laws of Logarithms
 Equation in the form of index and logarithm:

$$N=a^x \iff \log_aN=x$$

 where $$a \gt0,\, a \ne1$$. Logarithms:

\begin{aligned} &\bullet \log_aa^x=x \\\\ &\bullet \log_a1=0 \\\\ &\bullet \log_aa=1 \end{aligned}

 The diagram shows the graphs of exponential and logarithmic functions.

 We can see that 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. The logarithms of negative numbers and of zero are undefined. Law of logarithms:

\begin{aligned} &\bullet \log_axy=\log_ax+\log_ay \\\\ &\bullet \log_a\dfrac{x}{y}=\log_ax-\log_ay\\\\ &\bullet \log_ax^n=n \log_ax \end{aligned}

 for any real number $$n$$ where $$a,x$$ and $$y$$ are positive numbers and $$a\ne1$$. Change of base of logarithms:

\begin{aligned} &\bullet \log_ab=\dfrac{\log_cb}{\log_ca} \\\\ &\bullet \log_ab=\dfrac{1}{\log_ba} \end{aligned}

 where $$a$$, $$b$$ and $$c$$ are positive numbers, $$a \ne 1$$ and $$c \ne 1$$. $$\lg=\log_{10}$$ (common logarithms) and $$\ln=\log_{e}$$ (natural logarithms) where $$e$$ is a constant.