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[tex]$10 \cdot 2^{\frac{3 t}{5}}=1000$[/tex]
What is the solution of the equation?
Round your answer, if necessary, to the nearest thousandth
[tex]$t \approx \square$[/tex]
Answer (2)
Divide both sides of the equation by 10: 2 5 3 t = 100 .
Take the logarithm base 2 of both sides: 5 3 t = lo g 2 100 .
Multiply both sides by 3 5 : t = 3 5 lo g 2 100 .
Calculate the value of t and round to the nearest thousandth: t ≈ 11.073 .
Explanation
Problem Setup We are given the equation 10 c d o t 2 5 3 t = 1000 and we want to solve for t , rounding to the nearest thousandth.
Isolating the Exponential Term First, we divide both sides of the equation by 10 to isolate the exponential term: 10 10 c d o t 2 5 3 t = 10 1000 2 5 3 t = 100
Taking the Logarithm Next, we take the logarithm base 2 of both sides of the equation: lo g 2 ( 2 5 3 t ) = lo g 2 ( 100 ) 5 3 t = lo g 2 ( 100 )
Solving for t Now, we multiply both sides by 3 5 to solve for t :
t = 3 5 lo g 2 ( 100 )
Change of Base Formula We can use the change of base formula to convert the logarithm to a more common base, such as the natural logarithm (base e ) or the common logarithm (base 10). Using the natural logarithm, we have: t = 3 5 ln ( 2 ) ln ( 100 )
Calculating t Now, we can calculate the value of t :
t = 3 5 ln ( 2 ) ln ( 100 ) ≈ 3 5 ⋅ 0.69315 4.60517 ≈ 3 5 c d o t 6.64386 ≈ 11.0731
Final Answer Rounding to the nearest thousandth, we get: t ≈ 11.073
Examples
Exponential equations like this one are used in various fields, such as finance, to model compound interest. For example, if you invest $10 at an annual interest rate such that it grows to $1000 over a certain period, this equation can help you determine the time it takes for the investment to reach that value. Understanding exponential growth is crucial for making informed financial decisions and planning for the future. This also applies to population growth, radioactive decay, and other phenomena that exhibit exponential behavior.
To solve 10 ⋅ 2 5 3 t = 1000 , first isolate the exponential term to get 2 5 3 t = 100 . Then, take the logarithm base 2 of both sides and solve for t to find that t ≈ 11.073 .
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