Solve the following exponential equation without using logarithms.
[tex]4^{9 x-7}=256[/tex]
The solution is [tex]x=[/tex] $\square$ .
(Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
Answer (2)
To solve 4 9 x − 7 = 256 , we express 256 as 4 4 and equate the exponents, leading to 9 x − 7 = 4 . Solving this gives x = 9 11 . Therefore, the solution is x = 9 11 .
;
Express 256 as a power of 4: 256 = 4 4 .
Rewrite the equation: 4 9 x − 7 = 4 4 .
Equate the exponents: 9 x − 7 = 4 .
Solve for x : x = 9 11 .
9 11
Explanation
Understanding the Problem We are given the exponential equation 4 9 x − 7 = 256 . Our goal is to solve for x without using logarithms.
Expressing 256 as a Power of 4 First, we express 256 as a power of 4. Since 4 4 = 256 , we can rewrite the equation as 4 9 x − 7 = 4 4 .
Equating the Exponents Now that the bases are equal, we can equate the exponents: 9 x − 7 = 4 .
Solving for x Next, we solve the linear equation 9 x − 7 = 4 for x . Add 7 to both sides of the equation: 9 x − 7 + 7 = 4 + 7 , which simplifies to 9 x = 11 .
Isolating x Finally, divide both sides by 9 to isolate x : 9 9 x = 9 11 , which gives us x = 9 11 .
Final Answer Therefore, the solution to the exponential equation 4 9 x − 7 = 256 is x = 9 11 .
Examples
Exponential equations are used in various real-world applications, such as modeling population growth, radioactive decay, and compound interest. For example, if a population doubles every year, the population size can be modeled by an exponential equation. Solving such equations helps us predict future population sizes or determine the time it takes for a population to reach a certain size. Similarly, in finance, exponential equations are used to calculate the future value of an investment with compound interest. Understanding how to solve exponential equations is crucial for making informed decisions in these areas.
