Question

(1) Simplify Log2[27/4]-Log2[9/8]+Log2[1/24]. (2)What is Log7[49a]-Log10[0.01]? (3) Solve this: 9e2x-30ex+25=0 (4) If 81-2x=1/4x, find x

Answer 1

Given

The equation is given as

$9e^{2x}-30e^x+25=0$

Explanation

Solve and simplify the expression

$9e^{2x}-30e^x+25=0$

Let substitute

$e^x=u$

in the equation.

$9u^2-30u+25=0$

Solve the quadratic equation,

$\begin{gathered} u=\frac{30\pm\sqrt{30^2-4\times9\times25}}{2\times9} \\ u=\frac{30\operatorname{\pm}\sqrt{900-900}}{18} \\ u=\frac{30}{18}=\frac{5}{3} \end{gathered}$

So the solution is

$\begin{gathered} (u-\frac{5}{3})^2=0 \\ u=\frac{5}{3} \end{gathered}$

Now find the value of x ,

$\begin{gathered} e^x=\frac{5}{3} \\ x=ln\text{ }\frac{5}{3} \end{gathered}$

Then the value of x is obtained as

$x=0.51$

Answer

Hence the solution of the equation is 0.51.