Question

How do you solve ${{\left( \dfrac{3}{4} \right)}^{x}}=\dfrac{27}{64}$ ?

Answer 1

Hint: For this problem we prefer solving it by converting the numbers into real number indices as solving it by logarithm will require logarithmic tables or calculators. First, we will convert the fraction $\dfrac{27}{64}$ into a power of $\dfrac{3}{4}$ and then we can compare it to the left-hand side of the problem to find the value of $x$ .

Complete step by step answer:
This problem can be solved either by using logarithm or by just simply converting the fraction $\dfrac{27}{64}$ into a fraction having real number indices. As the first method requires logarithmic tables and calculators, we will solve it by following the second method.
If ${{\left( \dfrac{a}{b} \right)}^{x}}=\dfrac{p}{q}$ then for the solution we have to convert the fraction $\dfrac{p}{q}$ to some power of $\dfrac{a}{b}$
And if $\dfrac{p}{q}$ becomes ${{\left( \dfrac{a}{b} \right)}^{m}}$ , then we can simply compare with the left-hand side ${{\left( \dfrac{a}{b} \right)}^{x}}$ and get the solution $x=m$
So, in the fraction $\dfrac{27}{64}$ , we can break the numerator $27$ into its factor $3$ in this manner:
$27=3\times 3\times 3$
Now, we further simplify it by converting into real number indices as:
$27={{3}^{3}}$
Also, we take the number $64$ from the denominator of the fraction $\dfrac{27}{64}$ and break it into its factor $4$ as:
$64=4\times 4\times 4$
We can further simplify this by converting into real number indices as:
$64={{4}^{3}}$
Now, the fraction $\dfrac{27}{64}$ becomes $\dfrac{{{3}^{3}}}{{{4}^{3}}}$
And $\dfrac{{{3}^{3}}}{{{4}^{3}}}={{\left( \dfrac{3}{4} \right)}^{3}}$
Comparing it with the left-hand side of the problem, ${{\left( \dfrac{3}{4} \right)}^{x}}={{\left( \dfrac{3}{4} \right)}^{3}}$
Hence, $x=3$
Therefore, we conclude that the solution of the given equation is $x=3$.

Note:
While converting a fraction into a fraction of real number indices we must keep in mind that the numerator and the denominator have to have a common number as its power, which in this case is $3$ . This problem can also be solved by taking logarithms on both sides of the equation. Then, we get,
$\Rightarrow x\ln \left( \dfrac{3}{4} \right)=\ln \left( \dfrac{27}{64} \right)$
We can write $\dfrac{27}{64}$ as ${{\left( \dfrac{3}{4} \right)}^{3}}$ . Then,
$\Rightarrow x\ln \left( \dfrac{3}{4} \right)=3\ln \left( \dfrac{3}{4} \right)$
$\Rightarrow x=3$