Question

How do you solve ${{4}^{3x}}=512$?

Answer 1

Hint: We know that if ${{a}^{b}}$ is equal to c then b will be equal to ${{\log }_{a}}c$ and we know the property of logarithm that ${{\log }_{a}}{{b}^{x}}$ is equal to $x{{\log }_{a}}b$ we can use these property to solve the above question.

Complete step by step answer:
The given equation is ${{4}^{3x}}=512$
We can use logarithm here we know that ${{\log }_{a}}c$ = b implies ${{a}^{b}}$ is equal to c, so we can write $3x={{\log }_{4}}512$
Now we have to calculate the value of ${{\log }_{4}}512$ we can use log table to find $\dfrac{\ln 512}{\ln 4}$ but there is a simpler method we can see that 512 is equal to ${{2}^{9}}$ we can replace it in the equation
So we can write $3x={{\log }_{4}}{{2}^{9}}$
We know the property of logarithm that ${{\log }_{a}}{{b}^{x}}$ = $x{{\log }_{a}}b$
So the value of ${{\log }_{4}}{{2}^{9}}$ is equal to $9{{\log }_{4}}2$
Square root of 4 is equal to 2
So the value of ${{\log }_{4}}2$ is equal to $\dfrac{1}{2}$
Now we can write $3x=9\times \dfrac{1}{2}$
So $x=\dfrac{1}{3}\times 9\times \dfrac{1}{2}=\dfrac{3}{2}$

Note:
We can solve the above problem by another method. We can write 4 as square of 2
${{4}^{3x}}={{\left( {{2}^{2}} \right)}^{3x}}$ . We know that ${{\left( {{a}^{b}} \right)}^{c}}={{a}^{bc}}$ so applying this we get ${{4}^{3x}}={{2}^{6x}}$ . So then we can write
${{2}^{6x}}=512$ and if we observe we can see that 512 = ${{2}^{9}}$ . So we can write ${{2}^{6x}}={{2}^{9}}$
If a${{a}^{b}}={{a}^{c}}$ then the value of b is equal to the value of c so applying this we get
6x is equal to 9 that implies x is equal to $\dfrac{9}{6}$ . Further solving we can say x is equal 1.5