If $36^{12-m}=6^{2 m}$, what is the value of $m$?
Answer (1)
Rewrite the equation using the fact that 36 = 6 2 , so the equation becomes ( 6 2 ) 12 − m = 6 2 m .
Simplify the left side of the equation using the power of a power rule: 6 2 ( 12 − m ) = 6 2 m .
Since the bases are equal, equate the exponents: 2 ( 12 − m ) = 2 m .
Solve the resulting linear equation for m : m = 6 .
Explanation
Rewrite the equation We are given the equation 3 6 12 − m = 6 2 m and asked to find the value of m . To solve this, we need to express both sides of the equation with the same base. Since 36 = 6 2 , we can rewrite the equation in terms of base 6.
Apply the power of a power rule Substitute 36 with 6 2 in the given equation: ( 6 2 ) 12 − m = 6 2 m Using the power of a power rule, we get: 6 2 ( 12 − m ) = 6 2 m
Equate the exponents Since the bases are equal, we can equate the exponents: 2 ( 12 − m ) = 2 m
Solve for m Now, we solve the equation for m :
24 − 2 m = 2 m Add 2 m to both sides: 24 = 4 m Divide both sides by 4: m = 4 24 m = 6
Final Answer Therefore, the value of m is 6.
Examples
Understanding exponential equations is crucial in various fields, such as finance and computer science. For instance, calculating compound interest involves exponential growth. If you invest 1000 inana cco u n tt ha tp a ys 5 A = 1000(1.05)^t$. Solving for t if you want to reach a certain amount involves using logarithms, which are closely related to exponential functions. This type of problem helps in making informed financial decisions.
