Courses
Courses for Kids
Free study material
Offline Centres
More

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

Last updated date: 26th Feb 2024
Total views: 338.7k
Views today: 9.38k
Verified
338.7k+ views
Hint: Write $\left( \dfrac{27}{64} \right)$ in the R.H.S. as exponent of $\left( \dfrac{4}{3} \right)$ by using the formula: - $\dfrac{1}{{{a}^{m}}}={{a}^{-m}}$. Now, compare the bases on both the sides and equate the exponents to form a linear equation in x. Solve this equation for the value of x to get the answer.

Here, we have been provided with the exponential expression: - ${{\left( \dfrac{4}{3} \right)}^{x}}=\left( \dfrac{27}{64} \right)$ and we are asked to solve it. That means we have to find the value of x.
Now, we can write $\left( \dfrac{27}{64} \right)$ in the R.H.S. as exponential form with base $\dfrac{3}{4}$. Here, $\dfrac{27}{64}=\dfrac{{{3}^{3}}}{{{4}^{3}}}$. Using the identity $\dfrac{{{a}^{m}}}{{{b}^{m}}}={{\left( \dfrac{a}{b} \right)}^{m}}$, we get,
$\Rightarrow {{\left( \dfrac{4}{3} \right)}^{x}}={{\left( \dfrac{3}{4} \right)}^{3}}$
Using the formula: - $\dfrac{1}{{{a}^{m}}}={{a}^{-m}}$ in the R.H.S. to write ${{\left( \dfrac{3}{4} \right)}^{3}}={{\left( \dfrac{4}{3} \right)}^{-3}}$, we get,
$\Rightarrow {{\left( \dfrac{4}{3} \right)}^{x}}={{\left( \dfrac{4}{3} \right)}^{-3}}$
As we can see that both the sides of the above exponential expression contains $\left( \dfrac{4}{3} \right)$ as the base. So, we can equate the exponents by removing the base from both the sides. So, we have,
$\Rightarrow x=-3$
Note: One may note that here we have used some basic formulas of the topic ‘exponents and powers’ to solve the question. You must remember some basic formulas such as: - ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$, $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ and ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ because they are used with the help of logarithm. We can take log to the base $\dfrac{4}{3}$ or $\dfrac{3}{4}$, i.e., ${{\log }_{\dfrac{4}{3}}}$or ${{\log }_{\dfrac{3}{4}}}$, both the sides and use the property ${{\log }_{n}}n=1$ to get the answer. Here, n > 0 and $n\ne 1$.