Question - Solving an Exponential Equation with Base 2
Solution:
To solve the equation $$4^{3x} = 32^{x-5}$$, we can express both sides of the equation with a base of 2, as both 4 and 32 are powers of 2:$$4 = 2^2$$ and $$32 = 2^5$$.So, we can rewrite the equation as$$(2^2)^{3x} = (2^5)^{x-5}$$.Using the exponentiation rule $$ (a^b)^c = a^{bc} $$, we get$$2^{6x} = 2^{5x - 25}$$.Since the bases are the same, we can set the exponents equal to each other:$$6x = 5x - 25$$.Now, we solve for $$x$$:$$6x - 5x = -25$$$$x = -25$$.Thus, the solution to the equation is $$x = -25$$.
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