Question

Solving exponential equations with a common base problem in the picture!

Answer 1

Given:

The given expression is $18^{x^{2}+4 x+4}=18^{9 x+18}$

We need to determine the solution of the given expression.

Solution:

Let us solve the exponential equations with common base.

Applying the rule, if $a^{f(x)}=a^{g(x)}$ then $f(x)=g(x)$

Thus, we have;

$x^{2}+4 x+4=9 x+18$

Subtracting both sides of the equation by 9x, we get;

$x^{2}-5 x+4=18$

Subtracting both sides of the equation by 18, we have;

$x^{2}-5 x-14=0$

Factoring the equation, we get;

$x^2-7x+2x-14=0$

Grouping the terms, we have;

$(x^2-7x)+(2x-14)=0$

Taking out the common term from both the groups, we get;

$x(x-7)+2(x-7)=0$

Factoring out the common term (x - 7), we get;

$(x+2)(x-7)=0$

$x+2=0 \ and \ x-7=0$

   $x=-2 \ and \ x=7$

Thus, the solution of the exponential equations is x = -2 and x = 7.

Hence, Option C is the correct answer.