Logarithms

If an exponential equation is $a^{x}=y$, then $x$ is called logarithm of $y$ to the base of $a$ and is written as $\log_a(y)=x$. Here, $x = \text{Power / Exponent}$

$y = \text{Given number}$ $a = \text{Base}$

Example#1:

$3^{-4}=\frac{1}{81}$ this is an exponential form to write it in Logarithmic form simply use the expression $a^{x}=y$ $\leftrightarrow$ $\log_{a}{y}=x$

Here; $a=3, x= -4 \text{ and } y = \frac{1}{81}$

now put it in the other side

$\log_{3}({\frac{1}{81})}=-4$

Example#2:

Find the value of $\log_4(2)$

Solution:

• First let the log equal to a variable,

  $\log_4(2)=x$

• Convert it into exponential form

  $4^{x}=2$

• Solve the exponential form

  $2^{2x}=2$

• use the basic method of Base same powers equal

  $2x=1$

  $x=\frac{1}{2}$

• So the answer of $\log_4(2)=\frac{1}{2}$

Example#3:

Find the value of x if $\log_{64}(x)=\frac{-2}{3}$

Solution:

• Convert the log form into exponential form

  $64^{\frac{-2}{3}}=x$

• Solve the exponential form

  $(4^3)^{\frac{-2}{3}}=x$

  $(4)^{-2}=x$

  ${\frac{1}{16}}=x$

• So the value of $x$ in $\log_{64}(x)=\frac{-2}{3}$ is $\frac{1}{16}$