Answer
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Hint: We know that ${\log _e}a = k$and can also be written as $a = {e^k}$.
So we are going to use this formula to find the value of 4 from the given equation \[lo{g_4}7 = x\].
Then we substitute the value of ${4^2}$ in the equation \[lo{g_7}16\] for the value of 16 as $16 = {4^2}$ and find its value.
Complete step-by-step answer:
Given \[lo{g_4}7 = x\]……………………(1)
We have to find the value of \[lo{g_7}16\].
Now we know that ${\log _e}a = k$can also be written as $a = {e^k}$.
So by using the above equation we get from equation (1)
$ \Rightarrow lo{g_4}7 = x \Rightarrow 7 = {4^x}$
Now separating the 4 from the equation and taking x to the exponential of 7, we get
$ \Rightarrow 4 = {7^{\dfrac{1}{x}}}$
Now lets see the equation \[lo{g_7}16\]
Rules I used in solving the questions are mentioned below but we need to remember all the rules of logarithm to solve the questions correctly.
The base b logarithm of a number is the exponent that we need to raise the base in order to get the number.
We know that ${\log _e}{x^y} = y.{\log _b}x$, then
\[lo{g_7}16 = lo{g_7}{4^2} = 2lo{g_7}4\]
Now substituting the value of 4 in the above equation from equation (2), we get
\[ = 2lo{g_7}\left( {{7^{\dfrac{1}{x}}}} \right)\]
We know that ${\log _e}{x^y} = y.{\log _b}x$, then
\[ = 2\left( {\dfrac{1}{x}} \right)lo{g_7}7\]
We know that ${\log _e}e = 1$, then
\[ = \dfrac{2}{x}\]
If \[lo{g_4}7 = x\], then the value of \[lo{g_7}16\] will be \[\dfrac{2}{x}\]
So, option (D) is the correct answer.
Note: Throughout your study of algebra, you have come across many properties—such as the commutative, associative, and distributive properties. These properties help you take a complicated expression or equation and simplify it. The same is true with logarithms. There are a number of properties that will help you simplify complex logarithmic expressions. So, one must know these properties to get the desired solution.
So we are going to use this formula to find the value of 4 from the given equation \[lo{g_4}7 = x\].
Then we substitute the value of ${4^2}$ in the equation \[lo{g_7}16\] for the value of 16 as $16 = {4^2}$ and find its value.
Complete step-by-step answer:
Given \[lo{g_4}7 = x\]……………………(1)
We have to find the value of \[lo{g_7}16\].
Now we know that ${\log _e}a = k$can also be written as $a = {e^k}$.
So by using the above equation we get from equation (1)
$ \Rightarrow lo{g_4}7 = x \Rightarrow 7 = {4^x}$
Now separating the 4 from the equation and taking x to the exponential of 7, we get
$ \Rightarrow 4 = {7^{\dfrac{1}{x}}}$
Now lets see the equation \[lo{g_7}16\]
Rules I used in solving the questions are mentioned below but we need to remember all the rules of logarithm to solve the questions correctly.
The base b logarithm of a number is the exponent that we need to raise the base in order to get the number.
Logarithm power rule | logb(x y) = y ∙ logb(x) |
Logarithm of the base | logb(b) = 1 |
We know that ${\log _e}{x^y} = y.{\log _b}x$, then
\[lo{g_7}16 = lo{g_7}{4^2} = 2lo{g_7}4\]
Now substituting the value of 4 in the above equation from equation (2), we get
\[ = 2lo{g_7}\left( {{7^{\dfrac{1}{x}}}} \right)\]
We know that ${\log _e}{x^y} = y.{\log _b}x$, then
\[ = 2\left( {\dfrac{1}{x}} \right)lo{g_7}7\]
We know that ${\log _e}e = 1$, then
\[ = \dfrac{2}{x}\]
If \[lo{g_4}7 = x\], then the value of \[lo{g_7}16\] will be \[\dfrac{2}{x}\]
So, option (D) is the correct answer.
Note: Throughout your study of algebra, you have come across many properties—such as the commutative, associative, and distributive properties. These properties help you take a complicated expression or equation and simplify it. The same is true with logarithms. There are a number of properties that will help you simplify complex logarithmic expressions. So, one must know these properties to get the desired solution.
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