Question

Explain the following in exponential form.
i. ${\log _2}128 = 7$
ii. ${\log _{10}}0.0001 = - 4$
iii. ${\log _8}16 = \dfrac{4}{3}$

Answer 1

Hint:
Here, we will compare the general form of the logarithmic function and find the variable of the formula. Then we will use these variables in the formula of an exponential function to convert the given logarithmic function to exponential form.

Complete step by step solution:
i. ${\log _2}128 = 7$………………………..$\left( 1 \right)$
We will use the formula ${\log _a}b = x$
Comparing the above equation from ${\log _2}128 = 7$ we get
$a = 2 \\
  b = 128 \\
  x = 7 \\ $
As ${\log _a}b = x$ is also written as ${a^x} = b$.
We can write our equation by raising the power of a by x and keeping it equal to b as,
$ \Rightarrow {2^7} = 128$
As we can see the above value is true
So the exponential form of ${\log _2}128 = 7$is${2^7} = 128$.

ii. ${\log _{10}}0.0001 = - 4$ ………………………..$\left( 2 \right)$
We will use the formula ${\log _a}b = x$
Comparing the above equation from ${\log _{10}}0.0001 = - 4$, we get
$a = 10 \\
  b = 0.0001 \\
  x = - 4 \\$
As ${\log _a}b = x$ is also written as ${a^x} = b$ we can write our equation ………………………..$\left( 1 \right)$ by raising the power of a by $x$ and keeping it equal to b as,
$ \Rightarrow {10^{ - 4}} = 0.0001$
As we can see the above value is true
So exponential form of ${\log _{10}}0.0001 = - 4$is ${10^{{{10}^{ - 4}}}} = - 4$

iii. ${\log _8}16 = \dfrac{4}{3} - - - \left( 3 \right)$
We will use the formula ${\log _a}b = x$
Comparing the above equation from ${\log _8}16 = \dfrac{4}{3}$ we get
$ a = 8 \\
  b = 16 \\
  x = \dfrac{4}{3} \\ $
As ${\log _a}b = x$ is also written as ${a^x} = b$ we can write our equation (3) by raising the power of a by x and keeping it equal to b as,
$ \Rightarrow {8^{\dfrac{4}{3}}} = 16$
As we can write $8 = {2^3}$
$ \Rightarrow {\left( 2 \right)^{3 \times \dfrac{4}{3}}} = {2^4} = 16$
As we can see the above value is true

So the exponential form of ${\log _8}16 = \dfrac{4}{3}$ is ${8^{\dfrac{4}{3}}} = 16$.

Note:
Logarithm function is the opposite of exponential in the same way as subtraction is the opposite of addition. We know that Log sign undo exponentials and therefore another word for exponents is power. An exponential function is when a number is raised to a particular power whereas the logarithm is the exponent that a base has to be raised to make that number.