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Solving equations with the variable in an exponent often involves use of logarithms. An exception to this is when both sides of the equation can be expressed as powers of the same number.
Since , we can replace the right side with a power of 5:
We now have an exponential equation where each side of the equation is a power of the same number. In this situation we do not need logarithms.
This equation tells use that 5 to the (2x-7) power results in the same number as 5 to the -3 power. The only way this can be is if the two exponents are equal. So
2x-7 = -3
This is a very simple equation to solve:
2x = 4
x = 2
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