Question

How do you express ${\log _3}8$ in terms of common logarithms ?

Answer 1

In this question, we need to express ${\log _3}8$ in terms of common logarithmic function. Here we will use the concept of logarithm and the properties of logarithm. We know that $x = {b^y}$ is an exponential function and the inverse of this function is $y = {\log _b}x$ which is a logarithmic function. Since in this question we are given a logarithmic function, we convert it into exponential function using the rule explained. Then we take $\log$ on both sides and simplify further to get the desired result.

Complete step-by-step solution:
Given the logarithmic function ${\log _3}8$
We are asked to express the given expression in the equation (1) in terms of common logarithms.
We know that logarithm is the inverse of exponential function. Note that logarithm form and index (exponential) 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.
Using the concept mentioned above, we can solve the given expression and then simplify it further.
Let us take the given expression as, $y = {\log _3}8$ …… (2)
Now comparing it with the general form given in the equation (1).
We have here $b = 3$ and $x = 8$
Thus comparing we get,
${3^y} = 8$
Now taking common $\log$(to the base 10) on both sides, we get,
$\Rightarrow {\log _{10}}({3^y}) = {\log _{10}}8$
$\Rightarrow {\log_{10}}{3^y}= {\log _{10}}8$ …… (3)
Now consider the term $\log {3^y}$.
We know that $\log {a^n} = n\log a$
Here $a = 3$ and $n = y$
Hence we get, $\log {3^y} = y\log 3$
Now the equation (3) becomes,
$\Rightarrow y{\log _{10}}3 = lo{g_{10}}8$
$\Rightarrow y = \dfrac{{{{\log }_{10}}8}}{{{{\log }_{10}}3}}$
We have the change of base formula which is given by,
${\log _a}b = \dfrac{{\log b}}{{\log a}}$
Hence we get,
$\Rightarrow y = \dfrac{{\log 8}}{{\log 3}}$
$\Rightarrow y \approx \dfrac{{0.9031}}{{0.4771}}$
$\Rightarrow y \approx 1.8929$
Hence we express ${\log _3}8$ in terms of common logarithms as $1.8929$.

Note: If the question has the word log or $\ln$, it represents the given function as logarithmic function. Note that we have two types of logarithmic function.
One is a common logarithmic function which is represented as a log and its base is 10.
The other one is a natural logarithmic function represented as $\ln$ and its base is $e$.
We must know the basic properties of logarithmic functions and note that these properties hold for both log and $\ln$ functions.
Some properties of logarithmic functions are given below.
(1) $\log (x \cdot y) = \log x + \log y$
(2) $\log \left( {\dfrac{x}{y}} \right) = \log x - \log y$
(3) $\log {x^n} = n\log x$
(4) $\log 1 = 0$
(5) ${\log _e}e = 1$