$3.4^x+\frac{9^{x+2}}{3}=6.4^{x+1}-\frac{9^{x+1}}{2}$
Answer (1)
Rewrite the equation in terms of powers of 2 and 3.
Simplify the equation using exponent rules.
Combine like terms and isolate the exponential terms.
Solve for x by taking the logarithm of both sides: − 2 1 .
Explanation
Problem Analysis We are given the equation 3. 4 x + 3 9 x + 2 = 6. 4 x + 1 − 2 9 x + 1 . Our goal is to solve for x .
Rewrite with Powers of 2 and 3 First, let's rewrite the equation in terms of powers of 2 and 3. Since 4 = 2 2 and 9 = 3 2 , we have: 3 ( 2 2 ) x + 3 ( 3 2 ) x + 2 = 6 ( 2 2 ) x + 1 − 2 ( 3 2 ) x + 1
Simplify Exponents Now, simplify the equation using exponent rules: 3 ( 2 2 x ) + 3 3 2 x + 4 = 6 ( 2 2 x + 2 ) − 2 3 2 x + 2
Further Simplification Further simplification gives us: 3 ( 2 2 x ) + 3 2 x + 3 = 6 ( 4 ) ( 2 2 x ) − 2 1 ( 3 2 x + 2 ) 3 ( 2 2 x ) + 27 ( 3 2 x ) = 24 ( 2 2 x ) − 2 9 ( 3 2 x )
Combine Like Terms Combine like terms by moving terms with 2 2 x and 3 2 x to opposite sides of the equation: 27 ( 3 2 x ) + 2 9 ( 3 2 x ) = 24 ( 2 2 x ) − 3 ( 2 2 x )
Simplify Simplify the equation: 2 63 ( 3 2 x ) = 21 ( 2 2 x )
Divide by 21 Divide both sides by 21: 2 3 ( 3 2 x ) = 2 2 x
Rewrite Rewrite the equation: 3 ( 3 2 x ) = 2 ( 2 2 x ) 3 ( 9 x ) = 2 ( 4 x )
Divide by 4 x Divide both sides by 4 x :
3 ( 4 9 ) x = 2
Divide by 3 Divide both sides by 3: ( 4 9 ) x = 3 2
Take the Logarithm Take the logarithm of both sides: x ln ( 4 9 ) = ln ( 3 2 )
Solve for x Solve for x: x = l n ( 4 9 ) l n ( 3 2 )
Simplify Simplify the expression: x = l n ( 9/4 ) l n ( 2/3 ) = 2 l n ( 3 ) − 2 l n ( 2 ) l n ( 2 ) − l n ( 3 ) = 2 ( l n ( 3 ) − l n ( 2 )) l n ( 2 ) − l n ( 3 ) = − 2 1
Final Answer Therefore, the solution for x is − 2 1 .
Examples
Exponential equations are used in various fields such as finance, biology, and physics. For instance, they can model population growth, radioactive decay, and compound interest. Understanding how to solve exponential equations allows us to predict future values, determine decay rates, and make informed decisions based on mathematical models. In finance, it helps in calculating investment growth, while in biology, it aids in understanding bacterial growth or drug decay in the body. These equations are fundamental in quantitative analysis and forecasting.
