Question

How do i solve this exponential equation​

Answer 1

Answer:

  x = 0 or 1/2

Step-by-step explanation:

Solving an exponential equation of this nature with different bases and coefficients is always a bit of an ad hoc affair. Here, we can divide by the right side expression and make a substitution that turns it into a quadratic.

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  $3\cdot16^x+2\cdot81^x=5\cdot36^x\\\\0.6\left(\dfrac{4}{9}\right)^x+0.4\left(\dfrac{9}{4}\right)^x=1\\\\\textsf{Let z=\left(\dfrac{4}{9}\right)^x }\\\\0.6z+\dfrac{0.4}{z}=1\qquad\text{after substitution}\\\\3z^2-5z +2 = 0\qquad\text{multiply by 5z and put in standard form}\\\\(3z-2)(z-1)=0\qquad\text{factor}\\\\z=\dfrac{2}{3}\quad\text{or}\quad z=1\\\\x=\dfrac{\log(z)}{\log(4/9)}=\boxed{\dfrac{1}{2}\quad\text{or}\quad 0}$

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Additional comment

A graphing calculator often solves these very nicely, especially when the equation is put into the form f(x) = 0.