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How do you rewrite \[{\log _7}343 = 3\] in exponential form?

Last updated date: 20th Jun 2024
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Hint: The given function is the logarithm function it can be defined as logarithmic functions are the inverses of exponential functions. By using one of the Basic Properties of logarithmic that is \[{\log _b}\left( x \right) = y\] we can write the given function as an exponential form.

Complete step-by-step solution:
The function from positive real numbers to real numbers to real numbers is defined as \[{\log _b}:{R^ + } \to R \Rightarrow {\log _b}\left( x \right) = y\], if \[{b^y} = x\], is called logarithmic function or the logarithm function is the inverse form of exponential function.
Exponential notation is an alternative method of expressing numbers. Exponential numbers take the form an, where a is multiplied by itself n times. In exponential notation, a is termed the base while n is termed the power or exponent or index
The definition of a logarithm shows an equation written in logarithmic form \[y = {\log _b}\left( x \right)\], and the same equation written in exponential form, \[{b^y} = x\]. Let’s compare the two equations and look at what is the same and what is different. In both equations the y stays on the left side and the x stays on the right side, the only thing that moved was the b called the “base”. Identifying and moving the base is the key to changing from logarithmic form into exponential form.
Now, Consider the logarithm function
 \[ \Rightarrow \,\,\,{\log _7}343 = 3\]

In this function, the base of the logarithmic equation is 7 and as the base moved from the left side of the equation to the right side of the equation the number 3 moved up and became the exponent, creating an exponential equation. The 3 and the 343 did not change sides and the word “log” was dropped.
\[i.e.,\,\,{\log _7}343 = 3\]
\[\therefore \,\,\,\,343 = {7^3}\]

Note: The logarithmic equation can be converted to the exponential form. The exponential form of a number is defined as the number of times the number is multiplied by itself. The general form of logarithmic equation is \[{\log _x}y = b\] and it is converted to exponential form as \[y = {x^b}\]. Hence, we obtain the result or solution for the equation.