Question

How do you calculate ${\log _9}14$ with a calculator?

Answer 1

Hint: We are asked to calculate ${\log _9}14$ with the help of a calculator. To do this, let us first simplify it and then use the calculator. We make use of the fact that logarithm form and index form are interchangeable. i.e. ${\log _b}x = y$ implies ${b^y} = x.$ Then we apply log on both sides to the index form. And we try to get the expression for y. Then calculate the value using the calculator.

Complete step by step solution:
We are given a function ${\log _9}14$
We need to calculate the value of ${\log _9}14$ with the help of a calculator.
We know that logarithm is the inverse of exponential function. Note that logarithm form and index form are interchangeable.
i.e. if ${b^y} = x$, then we have ${\log _b}x = y$ …… (1)
Where log denotes the logarithmic function.
Here x is an argument of logarithm function which is always positive.
And b is called the base of the logarithm function.
Now we interchange ${\log _9}14$ to index form using equation (1).
Here $b = 9$ and $x = 14.$
Let ${\log _9}14 = y$.
Then by equation (1), we get,
$ \Rightarrow {9^y} = 14$
As y is in the index form, we take log on both sides we get,
$ \Rightarrow \log ({9^y}) = \log (14)$ ……(2)
We have the logarithm identity $\log {m^n} = n\log m$
Here $m = 9$ and $n = y$.
Hence the equation (2) becomes,
$ \Rightarrow y\log 9 = \log 14$
Now we isolate the variable y.
Taking $\log 9$ to the right hand side, we get,
$ \Rightarrow y = \dfrac{{\log 14}}{{\log 9}}$
Now we use the calculator to find the value of y.
In the calculator put $\log 14$ and then divide sign $ \div $ and now put $\log 9$. Then press the equal sign $ = $.
We will obtain the answer as 1.201.
$\therefore y = 1.201$.

Hence the value of ${\log _9}14$ is 1.201.

Note :
We must make use of a scientific or graphical calculator to find the value of ${\log _9}14$.
To calculate ${\log _9}14$, we can make use of change of base law.
According to this law, ${\log _a}b = \dfrac{{{{\log }_c}b}}{{{{\log }_c}a}}$
Where c can be any value. We take here the base $c = 10$.
In the above problem ${\log _9}14$, we have $a = 9$ and $b = 14$.
Now using change of base law we get,
${\log _9}14 = \dfrac{{{{\log }_{10}}14}}{{{{\log }_{10}}9}}$
Now using the calculator find the value of ${\log _{10}}14$ and ${\log _{10}}9$. Then divide the answer.
Hence we get the value of ${\log _9}14$ as 1.201.