Solve the following equation.
[tex]3(11^{2x}) = 46,887[/tex]
[tex]x \approx[/tex]
(Do not round until the final answer. Then round to the nearest whole number as needed.)
Answer (1)
Divide both sides of the equation by 3: 1 1 2 x = 15629 .
Take the logarithm base 11 of both sides: 2 x = lo g 11 ( 15629 ) .
Use the change of base formula: x = 2 l o g 10 ( 11 ) l o g 10 ( 15629 ) .
Calculate and round to the nearest whole number: x ≈ 2 .
Explanation
Problem Analysis We are given the equation 3 ( 1 1 2 x ) = 46887 and we need to solve for x , rounding the final answer to the nearest whole number.
Isolating the Exponential Term First, we divide both sides of the equation by 3 to isolate the exponential term: 3 3 ( 1 1 2 x ) = 3 46887 1 1 2 x = 15629
Applying Logarithms Next, we take the logarithm base 11 of both sides of the equation: lo g 11 ( 1 1 2 x ) = lo g 11 ( 15629 ) Using the property of logarithms, we have: 2 x = lo g 11 ( 15629 )
Solving for x Now, we divide both sides by 2 to solve for x :
x = 2 lo g 11 ( 15629 )
Change of Base Formula We can use the change of base formula to convert the logarithm to base 10: x = 2 l o g 10 ( 11 ) l o g 10 ( 15629 )
Calculating the Value of x Now, we calculate the value of x :
x = 2 l o g 10 ( 11 ) l o g 10 ( 15629 ) ≈ 2 1.0414 4.1938 ≈ 2 4.027 ≈ 2.0136
Rounding to the Nearest Whole Number Finally, we round the value of x to the nearest whole number: x ≈ 2
Examples
Exponential equations are used in various real-world applications, such as calculating population growth, radioactive decay, and compound interest. For example, if you invest money in an account with compound interest, the amount of money you have after a certain time can be modeled using an exponential equation. Solving these equations helps you determine how long it will take for your investment to reach a certain value, or how much interest you need to earn to reach your financial goals. Understanding exponential equations is crucial for making informed decisions in finance, science, and many other fields.
