Question

How do i solve this exponential equation
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Answer 1

x = 0 or 1/2

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((4)/(9)\right)^x+0.4\left((9)/(4)\right)^x=1\\\\\textsf{Let $z=\left((4)/(9)\right)^x$}\\\\0.6z+(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=(2)/(3)\quad\text{or}\quad z=1\\\\x=(\log(z))/(\log(4/9))=\boxed{(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.