Question

What is the exact value of $x$ in ${{5.21}^{5x}}=16$?

Answer 1

Hint: In the given question as we can see clearly some basic concepts of exponents and power are required with logarithmic properties involved or some simplifications are done. We basically have to solve the given equation using logarithmic properties.

Complete step by step solution:
In the given question, we are given an equation ${{5.21}^{5x}}=16$, which we need to solve for x while using different sections of mathematics of which few are stated in hint.
So now, we have a product of 5 with 21 which is raised to power 5x on the left-hand side of the equation and is set equivalent to 16 on the right-hand side.
Given: ${{5.21}^{5x}}=16$
Now we will divide both the left hand side and the right hand side by 5 and then we get,
$\Rightarrow \dfrac{{{5.21}^{5x}}}{5}=\dfrac{16}{5}$ and now cancelling 5 from the left hand side in order to simplify we get, ${{21}^{5x}}=\dfrac{16}{5}$
Now writing the right hand side of the attained equation in decimal form we get $\dfrac{16}{5}=3.2$ .
Now, the equation becomes ${{21}^{5x}}=3.2$
Taking the transpose of the equation and and log on both sides of the equation we get:
$\log (3.2)=\log ({{21}^{5x}})$
Now using the log properties that $\log {{m}^{n}}=n\log m$ we will get the above equation as $\log 3.2=5x\log (21)$.
Now in order to find the value of x what we need to do is we need to divide by some number so that we are left with x variable on one side, so for that we will divide the equation right hand side and left hand side with $5log(21)$ .
So now we will get the equation as $x=\dfrac{\log 3.2}{5\log 21}$ .

Note: We must be aware and careful while using the logarithmic properties and should do the calculations carefully because in these questions there are chances of mistakes which may occur very easily .