Question

Suppose that a and b are positive real numbers such that log27a + log9b = 7/2 and log27b + log9a = 2/3. FInd the value of ab.

Answer 1

Answer:243

Step-by-step explanation:

Hope you found it useful. ....

Answer 2

Concept Introduction:-

Logarithm is a amount representing the electricity to which a hard and fast number (the base) need to be raised to supply a given number.

Given Information:-

We have been given that $log_{27}a + log_{9}b =\frac{7}{2}$and $log_{27}b + log_{9}a = \frac{2}{3}$

To Find:-

We have to find that the value of $ab$.

Solution:-

According to the problem

Understand that $\log _{27} a+\log _{9} b=\frac{7}{2}$ and $\log _{27} b+\log _{9} a=\frac{2}{3}\\\Rightarrow \frac{1}{3} \log _{3} a+\frac{1}{2} \log _{3} b=\frac{7}{2} \ldots(1)$

and $\frac{1}{3} log _{3} b+\frac{1}{2} \log _{3} a=\frac{2}{3} \ldots (2)$

Add equation $(1)$ and $(2)$,

$\begin{aligned}&\Rightarrow \frac{1}{3} \log _{3}(a b)+\frac{1}{2} \log _{3}(a b)=\frac{7}{2}+\frac{2}{3}=\frac{25}{6} \\&\frac{5}{6} \log _{3}(a b)=\frac{25}{6} \\&\log _{3}(a b)=5 \\&a b=3^{5}=243\end{aligned}$

Final Answer:-

The value of $ab$ is $243$.

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