How do you solve 4^(x - 1) = 3^(2x), without using a calculator and using log?

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We have to solve the equation 4^(x - 1) = 3^2x

We see that the base of the two terms cannot be made the same, so we cannot equate the exponents to determine x without using logarithms.

4^(x - 1) = 3^2x

take the log of both the sides

log [ 4^(x - 1)] = log [3^2x]

use the relation log a^b = b*log a

=> (x - 1)*log 4 = 2x*log 3

=> (x - 1)/2x = log 3 / log 4

=> 1/2 - 1/2x = log 3 / log 4

=> (1/2x) = (1/2) - (log 3 / log 4)

=> x = 1/ [1 - 2*(log 3/ log 4)]

**The required value of x is x = 1/ [1 - 2*(log 3/ log 4)]**

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