Rewrite the following equations in the logarithm from: ${{5}^{0}}=1$
Answer
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Hint: The exponential if of the form ${{b}^{y}}=x$. Convert it to logarithmic form $y={{\log }_{b}}x$, where b is the base of the number and b should be greater than zero (b> 0).
Complete step-by-step answer:
Logarithms were created to be the inverse of exponential function. To give us the ability to solve the problem $x={{b}^{y}}$ for y.
For $x>0$ and $b>0,b\ne 1,y={{\log }_{b}}x$ is equivalent to ${{b}^{y}}=x$.
The definition of logarithm gives us the ability to write an equation two different ways and even though these two equations look different, both equations have the same meaning.
From the definition of a logarithm we get two equations $y={{\log }_{b}}x$and ${{b}^{y}}=x$. The equations $y={{\log }_{b}}x$and ${{b}^{y}}=x$ are different ways of expressing the same thing. The equation $y={{\log }_{b}}x$ is written in logarithmic form and the equation ${{b}^{y}}=x$is written in exponential form. The two equations are just different ways of writing the same thing.
To change from exponential form to logarithmic form, identify the base of the exponential equation and move the base to the other side of the equal sign and add the word “log”. Do not move anything but the base, the other number or variables will not change sides.
Here given ${{5}^{0}}=1$ of form ${{b}^{y}}=x$ [exponential]
$\begin{align}
& y={{\log }_{b}}x\Rightarrow 0={{\log }_{5}}x \\
& 0={{\log }_{5}}1 \\
& \Rightarrow {{\log }_{5}}1=0 \\
\end{align}$
In the exponential form, the number 5 in the equation is called the base, the same base as in the logarithmic form of equation “log base 5”.
Note: Do not spend too much time trying to understand the meaning of the different equations. All you need to know is that a logarithmic function is the inverse of an exponential function and all exponential equations can be written in logarithmic form.
Complete step-by-step answer:
Logarithms were created to be the inverse of exponential function. To give us the ability to solve the problem $x={{b}^{y}}$ for y.
For $x>0$ and $b>0,b\ne 1,y={{\log }_{b}}x$ is equivalent to ${{b}^{y}}=x$.
The definition of logarithm gives us the ability to write an equation two different ways and even though these two equations look different, both equations have the same meaning.
From the definition of a logarithm we get two equations $y={{\log }_{b}}x$and ${{b}^{y}}=x$. The equations $y={{\log }_{b}}x$and ${{b}^{y}}=x$ are different ways of expressing the same thing. The equation $y={{\log }_{b}}x$ is written in logarithmic form and the equation ${{b}^{y}}=x$is written in exponential form. The two equations are just different ways of writing the same thing.
To change from exponential form to logarithmic form, identify the base of the exponential equation and move the base to the other side of the equal sign and add the word “log”. Do not move anything but the base, the other number or variables will not change sides.
Here given ${{5}^{0}}=1$ of form ${{b}^{y}}=x$ [exponential]
$\begin{align}
& y={{\log }_{b}}x\Rightarrow 0={{\log }_{5}}x \\
& 0={{\log }_{5}}1 \\
& \Rightarrow {{\log }_{5}}1=0 \\
\end{align}$
In the exponential form, the number 5 in the equation is called the base, the same base as in the logarithmic form of equation “log base 5”.
Note: Do not spend too much time trying to understand the meaning of the different equations. All you need to know is that a logarithmic function is the inverse of an exponential function and all exponential equations can be written in logarithmic form.
Last updated date: 16th Sep 2023
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