Answer
True.
The equation is ${{3}^{5x}}=7$.
Work Step by Step
Consider the equation.
${{3}^{5x}}=7$
Write the equivalent form of the expression.
If any expression is in the form of $M={{b}^{p}}$ , then it is equivalent to the ${{\log }_{b}}M=p$.
Here, $b=3,M=7,p=5x$ .
$\begin{align}
& {{3}^{5x}}=7 \\
& 5x={{\log }_{3}}7 \\
\end{align}$
Apply change of base rule.
$\begin{align}
& 5x=\frac{\log 7}{\log 3} \\
& x=\frac{1}{5}\left( \frac{\log 7}{\log 3} \right) \\
\end{align}$
The value of $x=\frac{1}{5}\left( \frac{\log 7}{\log 3} \right)$ can be calculated by using of using Ti-84 and is below:
Step1: Press ON key.
Step2: Press LOG key.
Step3: Enter the value 7.
Step4: Press “)” key.
Step5: Press “$\div $” key.
Step6: Press LOG key.
Step7: Enter the value 3.
Step8: Press “)” key.
Step9: Press “(” key.
Step10: Enter the value 1.
Step11: Press “$\div $” key.
Step12: Enter the value 5.
Step13: Press “)” key.
Step14: Press ENTER key.
The result obtained is $0.3542487498$.
Therefore, the value of x is $0.3542$.
Check,
Substitute $0.3542$ for x in the given expression ${{3}^{5x}}=7$.
${{3}^{5\left( 0.3542 \right)}}\overset{?}{\mathop{=}}\,7$
The value of ${{3}^{5\left( 0.3542 \right)}}$ can be calculated by using of using Ti-84 and is below:
Step1: Press ON key.
Step2: Enter the value 3.
Step3: Press “^” key.
Step4: Press “(” key.
Step5: Enter the value $5$.
Step6: Press “)” key.
Step7: Enter the value $0.3542$.
Step8: Press ENTER key.
The result obtained is $6.99812575\approx 7$.
Thus, ${{3}^{5\left( 0.3542 \right)}}\overset{?}{\mathop{=}}\,7$.
It is true.
