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How do you solve the exponential equation ${5^{3x}} = {25^{x - 4}}$?

Last updated date: 13th Jun 2024
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Hint:The above question is based on the concept of exponential equations. The main approach towards solving this type of question is to first equate the base i.e., on both sides, the base should be 5 in such a way that the power becomes equal and we can find the value of x.

Complete step by step solution:
For solving exponential equations without logarithms, we need to have equations which are comparable exponential expressions on either side of equal to sign so that we can compare the power with the common base and further solve.
So now the above given exponential equation is given as below,
${5^{3x}} = {25^{x - 4}}$
So, the first step we need to do is to we need to equate the base on either side equal to sign.
Since on the left-hand side we have the base as 5b and on the right-hand side we have base as 25.
Since 25 is a perfect square of 5 so we can write the base 5 with the power 2.
\[{5^{3x}} = {\left( 5 \right)^{2\left( {x - 4} \right)}}\]
Now further by multiplying the indices we get ,
\[{5^{3x}} = {\left( 5 \right)^{2x - 8}}\]
Since the bases are same therefore, we can write it as,
3x = 2x - 8 \\
x = - 8 \\
Therefore, we get the value of the variable as \[x = - 8\].

Note: An important thing to note is that there is also an alternative method to solve the above exponential equation. We can apply log on both the left hand and right-hand side of the equation and by applying log properties we will get the same value as above.