6.3 Exponential equations (in class)

Solve the following exponential equations. If you have two or more answers, write the smallest one.

(a) [math]2^{x^2-5}=16[/math]

(b) [math]\sqrt{3^{x^2}}=9^{x}[/math]

(c) [math]\dfrac{81^{3x+2}}{243^{-x}}=3^4[/math]

(d) [math]25^{x-0.5}-4\cdot5^{x-1}=1[/math]

(e) [math]8^x+4^{x+1}=2^x+4[/math]

(f) [math]2^{|x|}=\left(\dfrac{1}{\sqrt{2}}\right)^{(x-1)}[/math]

Solve the following inequalities

(a) [math]2^x\geq3-2^{-x+1}[/math]

(b) [math]\displaystyle{\left(\frac{1}{\pi}\right)^{3x-2}}-1\geq0[/math]

(c) [math]\dfrac{9^x}{3^x+6}\leq1[/math]

Consider an exponential equation [math]9(9^x+9^{-x})-91(3^x+3^{-x})+100=0[/math].

(a) Let [math]3^x+3^{-x}=A[/math]. Express [math]9^x+9^{-x}[/math] in terms of [math]A[/math].

(b) Use the AM-GM inequality to find the minimum of [math]A[/math].
AM-GM inequality: If [math]p[/math] and [math]q[/math] are positive, then [math]\dfrac{p+q}{2}\geq\sqrt{pq}[/math]

(c) Find the value of [math]A[/math].

(d) Hence, solve the exponential equation. Write your answer in ascending order.

Suppose [math]a, b[/math] are the solutions to the equation [math]9^x-5\cdot3^x+27=0[/math]. Find [math]a+b[/math].