For what value of [tex]$x$[/tex] does [tex]$4^x=\left(\frac{1}{8}\right)^{x+3} ?$[/tex]
Answer (1)
Express both sides of the equation with the same base: 4 x = ( 8 1 ) x + 3 becomes ( 2 2 ) x = ( 2 − 3 ) x + 3 .
Simplify the exponents: 2 2 x = 2 − 3 ( x + 3 ) .
Equate the exponents: 2 x = − 3 ( x + 3 ) .
Solve for x : x = − 5 9 .
The solution is − 5 9 .
Explanation
Problem Analysis We are given the equation 4 x = ( 8 1 ) x + 3 and asked to find the value of x that satisfies it. To solve this, we will express both sides of the equation with the same base, which will allow us to equate the exponents and solve for x .
Expressing with the Same Base First, we express both 4 and 8 1 as powers of 2. We know that 4 = 2 2 and 8 1 = 2 − 3 . Substituting these into the original equation, we get ( 2 2 ) x = ( 2 − 3 ) x + 3 .
Simplifying Exponents Next, we simplify the exponents using the power of a power rule, which states that ( a m ) n = a mn . Applying this rule, we have 2 2 x = 2 − 3 ( x + 3 ) .
Equating Exponents Since the bases are now equal, we can equate the exponents: 2 x = − 3 ( x + 3 ) . Now we solve this linear equation for x .
Solving for x Expanding the right side of the equation, we get 2 x = − 3 x − 9 . Adding 3 x to both sides, we have 5 x = − 9 . Dividing both sides by 5, we find x = − 5 9 .
Final Answer Therefore, the value of x that satisfies the equation is − 5 9 .
Examples
Exponential equations like this one are used in various fields, such as calculating the growth of bacteria or the decay of radioactive substances. For example, if you have a population of bacteria that doubles every hour, you can use an exponential equation to predict the population size after a certain number of hours. Similarly, in finance, exponential functions are used to calculate compound interest. Understanding how to solve exponential equations is crucial for making predictions and informed decisions in these areas.
