Question

(1) Solve the following exponential equation: Show your work. 342-12 = 6561 (2) Solve the following logarithmic equation: Show your work. logy (z? + 4) - log7 (x + 1) - 343

Answer 1

34 - 12 = 6561

Factor of 6561 = 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = $3^8$
Now,


34r-12

Here base is same then power will also same.
4x - 12 = 8
4x = 20
x = 5


2.

logy (z? + 4) - log; (x + 1) - 343



log7 (22+4) – log, (x + 1) 343

Apply ::$\log_ma-\log_mb = \log_m\frac{a}{b}$
$\log _7\frac{\left(x^2+4\right)}{x+1}=\log_77^{343}$
$\frac{\left(x^2+4\right)}{x+1}=7^{343}$
$x^2+4=7^{343}(x+1)$
From here we get 2 value of x, But none of them varify the solution beacuse the domain of logarthem funtion is positive x.
So, we can say there is no value of x, Or No solution


After solving the equation we get, the value of x as
$x=\frac{7^{343}+\sqrt{7^{686}-4\left(4-7^{343}\right)}}{2},\:x=\frac{7^{343}-\sqrt{7^{686}-4\left(4-7^{343}\right)}}{2}$
$\mathrm{Verify\:Solutions}:\quad\\ x=\frac{7^{343}+\sqrt{7^{686}-4\left(4-7^{343}\right)}}{2}\;\;\mathrm{False},\\ \:\space x=\frac{7^{343}-\sqrt{7^{686}-4\left(4-7^{343}\right)}}{2}\space\;\;\;\mathrm{False}$
$\mathrm{No\:Solution\:for}\:x\in \mathbb{R}$