Question

How do you solve ${{4}^{2x}}=6$ ?

Answer 1

Hint: To solve ${{4}^{2x}}=6$ , we have to take logarithm on both sides. Then we have to use the logarithmic rules like $\log \left( {{a}^{m}} \right)=m\log a$ and solve for x.

Complete step by step solution:
We have to solve ${{4}^{2x}}=6$ . We have to take logarithms on both sides. Then the given expression becomes
$\Rightarrow \log \left( {{4}^{2x}} \right)=\log 6$
We know that $\log \left( {{a}^{m}} \right)=m\log a$ . Let us apply this formula on the above equation. Then the above equation becomes
$\Rightarrow 2x\log 4=\log 6$
We know that $\log 4=0.602$ and $\log 6=0.778$ . Hence, the equation becomes
$\Rightarrow 2x\times 0.602=0.778$
Let us multiply 2 with 0.602. We will get
$\Rightarrow 1.204x=0.778$
Let us take 1.204 to the RHS. Then the above expression becomes
$\Rightarrow x=\dfrac{0.778}{1.204}$
Let us now solve the RHS. We will get
$\Rightarrow x=0.646$

Hence, the required answer is 0.646.

Note: Students must know all the logarithmic rules to solve the questions that have variables as the power of a number or variable. Whenever such questions are to be solved, we must apply logarithm on both the sides. We can also solve the given question in an alternate way. First, we have to take logarithm on both sides.
$\Rightarrow \log \left( {{4}^{2x}} \right)=\log 6$
We know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ . Hence, the above equation becomes
$\Rightarrow \log \left( {{\left( {{4}^{2}} \right)}^{x}} \right)=\log 6$
We know that ${{4}^{2}}=16$ . We can write the above equation as
$\Rightarrow \log \left( {{16}^{x}} \right)=\log 6$
Now, we can use the formula $\log \left( {{a}^{m}} \right)=m\log a$ . We can write the above equation as
$\Rightarrow x\log 16=\log 6$
Let us take $\log 16$ to the RHS. We will get
$\Rightarrow x=\dfrac{\log 6}{\log 16}$
We know that $\log 6=0.778$ and $\log 16=1.204$ . Hence, the above equation becomes
$\Rightarrow x=\dfrac{0.778}{1.204}$
Let us now solve the RHS.
$\Rightarrow x=0.646$