Question

If $f\left( x \right) = \log x$, $g\left( x \right) = {x^3}$, then $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right]$ equals
(a) $f\left[ {g\left( a \right) + g\left( b \right)} \right]$
(b) $3f\left( {ab} \right)$
(c) $g\left[ {f\left( {ab} \right)} \right]$
(d) $g\left[ {f\left( a \right) + f\left( b \right)} \right]$

Answer 1

Here, we have to find the value of the composite function. Composite functions are those functions where one function is formed after putting another function inside it. To solve the question we will simplify the function in terms of $a$ and $b$. Then, we will apply the rules of logarithms to simplify the expression further, and get the correct option as the answer.
Formula used: We will use the rules of logarithms $\log {x^n} = n\log x$ and $\log x + \log y = \log \left( {xy} \right)$.

Complete step by step solution:
We will use the rules of logarithms to simplify the expression.
First, we will find the value of $g\left( a \right)$ and $g\left( b \right)$.
Substituting $x = a$ in the function $g\left( x \right) = {x^3}$, we get
$\Rightarrow g\left( a \right) = {a^3}$
Substituting $x = b$ in the function $g\left( x \right) = {x^3}$, we get
$\Rightarrow g\left( b \right) = {b^3}$
Next, we will find the value of $f\left[ {g\left( a \right)} \right]$.
Substituting $x = g\left( a \right)$ in the function $f\left( x \right) = \log x$, we get
$\Rightarrow f\left[ {g\left( a \right)} \right] = \log \left[ {g\left( a \right)} \right]$
Substituting $g\left( a \right) = {a^3}$in the expression, we get
$\Rightarrow f\left[ {g\left( a \right)} \right] = \log {a^3}$
We know the rule of logarithms $\log {x^n} = n\log x$.
Substituting $x = a$ and $n = 3$ in the rule of logarithms $\log {x^n} = n\log x$, we get
$\Rightarrow \log {a^3} = 3\log a$
Substituting $\log {a^3} = 3\log a$ in the equation $f\left[ {g\left( a \right)} \right] = \log {a^3}$, we get
$\Rightarrow f\left[ {g\left( a \right)} \right] = 3\log a \ldots \ldots \ldots \left( 1 \right)$
Now, we will find the value of $f\left[ {g\left( b \right)} \right]$.
Substituting $x = g\left( b \right)$ in the function $f\left( x \right) = \log x$, we get
$\Rightarrow f\left[ {g\left( b \right)} \right] = \log \left[ {g\left( b \right)} \right]$
Substituting $g\left( b \right) = {b^3}$in the expression, we get
$\Rightarrow f\left[ {g\left( b \right)} \right] = \log {b^3}$
Substituting $x = b$ and $n = 3$ in the rule of logarithms $\log {x^n} = n\log x$, we get
$\Rightarrow \log {b^3} = 3\log b$
Substituting $\log {b^3} = 3\log b$ in the equation $f\left[ {g\left( b \right)} \right] = \log {b^3}$, we get
$\Rightarrow f\left[ {g\left( b \right)} \right] = 3\log b \ldots \ldots \ldots \left( 2 \right)$
Finally, we will add equation $\left( 1 \right)$ and equation $\left( 2 \right)$ to get the value of the expression.
Adding equation $\left( 1 \right)$ and equation $\left( 2 \right)$, we get
$\Rightarrow f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = 3\log a + 3\log b$
Factoring out 3 from the terms of the expression, we get
$\Rightarrow f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = 3\left( {\log a + \log b} \right)$
Now, we know the rule of logarithms $\log x + \log y = \log \left( {xy} \right)$. Here, the base of the terms must be equal.
Therefore, substituting $x = a$ and $y = b$ in the formula, we get
$\Rightarrow \log a + \log b = \log \left( {ab} \right)$
Substituting $\log a + \log b = \log \left( {ab} \right)$ in the equation $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = 3\left( {\log a + \log b} \right)$, we get
$\Rightarrow f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = 3\log \left( {ab} \right)$
Now, we can observe that since $f\left( x \right) = \log x$, therefore $f\left( {ab} \right) = \log \left( {ab} \right)$.
Substituting $\log \left( {ab} \right) = f\left( {ab} \right)$ in the equation, we get
$f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = 3f\left( {ab} \right)$
Therefore, the value of the expression $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right]$ is $3f\left( {ab} \right)$.

Thus, the correct option is option (b).

Note:
We can also solve this problem by simplifying the given options and comparing them to the simplified value $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = 3\log a + 3\log b$. However, we need to remember the rules of logarithms. A common mistake we can make while simplifying the options is to use $\log x + \log y = \log \left( {x + y} \right)$ which is not correct.