Solve $\frac{1}{4}\left(4^{x-1}\right)+3=12$
$x=$ (round your answer to two decimal places)
Answer (2)
To solve the equation 4 1 ( 4 x − 1 ) + 3 = 12 , we isolate the exponential term, eliminate the fraction, use logarithms, and perform calculations which give us the final answer x ≈ 3.59 .
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3.59
Explanation
Problem Setup We are given the equation 4 1 ( 4 x − 1 ) + 3 = 12 and we want to solve for x , rounding the answer to two decimal places.
Isolating the Exponential Term First, subtract 3 from both sides of the equation: 4 1 ( 4 x − 1 ) = 12 − 3 4 1 ( 4 x − 1 ) = 9
Simplifying the Equation Next, multiply both sides of the equation by 4: 4 x − 1 = 9 × 4 4 x − 1 = 36
Applying Logarithms Now, take the logarithm base 4 of both sides: lo g 4 ( 4 x − 1 ) = lo g 4 ( 36 ) x − 1 = lo g 4 ( 36 )
Change of Base Use the change of base formula to express the logarithm in terms of the natural logarithm (ln): x − 1 = ln ( 4 ) ln ( 36 )
Isolating x Add 1 to both sides to isolate x :
x = ln ( 4 ) ln ( 36 ) + 1
Calculating x Now, we calculate the value of x :
x = ln ( 4 ) ln ( 36 ) + 1 ≈ 1.3863 3.5835 + 1 ≈ 2.5850 + 1 ≈ 3.5850 Rounding to two decimal places, we get x ≈ 3.59 .
Final Answer Therefore, the solution to the equation, rounded to two decimal places, is x = 3.59 .
Examples
Exponential equations are used in various real-world applications, such as calculating population growth, modeling radioactive decay, and determining compound interest. For example, if you invest money in an account with compound interest, the amount of money you have after a certain period can be calculated using an exponential equation. Understanding how to solve these equations allows you to predict future values and make informed decisions.
