Question

How do you solve ${{4}^{x+2}}=20$ ?

Answer 1

In the given question, we are given an equation in which the left-hand side involves the exponential and the right-hand side is the value the left-hand side is equal to.
So, in this question we need to solve this equation for x and for this we need to make use of the exponential property and logarithmic properties in order to reduce the complexity of the given equation.

Complete step-by-step solution:
So, now we know the logarithmic property that $\log {{m}^{n}}=n\log m$ .
Now, taking log both sides we get $log{{4}^{x+2}}=\log 20$ , and now using the above property we get $\left( x+2 \right)\log 4=\log 20$.
Now, using one more log property which is $\log mn=\log m+\log n$ .
Now we can write $\left( x+2 \right)\log 4=\log 20$ as $\left( x+2 \right)\log 4=\log \left( 4\times 5 \right)$
And now applying the above-mentioned property we get $\left( x+2 \right)\log 4=\log 4+\log 5$ and now further solving this for x we get
$\begin{aligned} & \Rightarrow \left( x+2-1 \right)\log 4=\log 5 \\\ & \Rightarrow \left( x+1 \right)=\dfrac{\log 5}{\log 4} \\\ \end{aligned}$
And now in order to get value of x we need to subtract 1 from left-hand side and right-hand side and we get
$\begin{aligned} & x=\dfrac{\log 5}{\log 4}-1 \\\ & \Rightarrow x=\dfrac{\log 5-\log 4}{\log 4} \\\ & \Rightarrow x=\dfrac{\log \dfrac{5}{4}}{\log 4} \\\ \end{aligned}$
Therefore, this is the most simplified form of the value of x and also, we can write as $x=\dfrac{\log 5}{\log 4}-1$ or else we can write as $x=\dfrac{\log 5-\log 4}{\log 4}$.
Hence, the answer written in any of this form is correct and hence this is how we have attained the value of x from the given equation.

Note: In this question basically we must know how to use logarithmic function and a major mistake is in applying the properties where we forget where log must be present and where log function presence is not necessary which leads to a highly wrong approach.