Problem 10
Question
Solve for \(t\) using natural logarithms. $$10=6 e^{0.5 t}$$
Step-by-Step Solution
Verified Answer
The value of \(t\) is \(\frac{\ln\left(\frac{5}{3}\right)}{0.5}\).
1Step 1: Isolate the Exponential
Start with the equation \(10 = 6 e^{0.5 t}\). To isolate the exponential term \(e^{0.5 t}\), we need to divide both sides by 6. This gives us:\[\frac{10}{6} = e^{0.5 t}\]Simplify the fraction. This reduces to:\[\frac{5}{3} = e^{0.5 t}\]
2Step 2: Apply the Natural Logarithm
Take the natural logarithm (\(\ln\)) of both sides to cancel out the exponential function, using the property \(\ln(e^x) = x\):\[\ln\left(\frac{5}{3}\right) = \ln(e^{0.5 t})\]This simplifies to:\[\ln\left(\frac{5}{3}\right) = 0.5 t\]
3Step 3: Solve for t
Now, solve for \(t\) by dividing both sides by 0.5:\[t = \frac{\ln\left(\frac{5}{3}\right)}{0.5}\]Calculate the natural logarithm and perform the division to find the value of \(t\).
Key Concepts
Exponential EquationsIsolation of VariablesSolving Equations
Exponential Equations
Exponential equations are a type of mathematical equation where the variable is located in the exponent. In the equation \(10 = 6 e^{0.5 t}\), the exponential term \(e^{0.5 t}\) contains the variable \(t\). Exponential equations typically feature a constant base raised to a variable power. These types of equations are frequently used in various fields such as science and finance.
To solve exponential equations, one common method involves isolating the exponential term, which allows us to apply logarithms for simplification. Knowing the properties and behavior of exponential functions, such as growth and decay, can provide insights into solving these equations efficiently.
To solve exponential equations, one common method involves isolating the exponential term, which allows us to apply logarithms for simplification. Knowing the properties and behavior of exponential functions, such as growth and decay, can provide insights into solving these equations efficiently.
Isolation of Variables
Isolation of variables is a crucial step in solving equations, especially exponential ones. The process involves rearranging the equation so that the variable of interest stands alone on one side of the equation. In our original problem \(10 = 6 e^{0.5 t}\), this meant isolating the exponential component \(e^{0.5 t}\).
To isolate the variable, start by performing inverse operations. Here, dividing both sides by 6 helps separate the exponential term: \(\frac{10}{6} = e^{0.5 t}\). Simplifying the fraction provides \(\frac{5}{3} = e^{0.5 t}\).
Isolation is key because it simplifies the process of applying further mathematical operations, like taking logarithms to solve for the variable. Ensuring the variable is isolated makes it easier to compare and manipulate both sides of the equation.
To isolate the variable, start by performing inverse operations. Here, dividing both sides by 6 helps separate the exponential term: \(\frac{10}{6} = e^{0.5 t}\). Simplifying the fraction provides \(\frac{5}{3} = e^{0.5 t}\).
Isolation is key because it simplifies the process of applying further mathematical operations, like taking logarithms to solve for the variable. Ensuring the variable is isolated makes it easier to compare and manipulate both sides of the equation.
Solving Equations
Solving equations, particularly those involving exponential functions, often requires the application of natural logarithms. In the step-by-step solution to our problem, after isolating the variable, the next move is to take the natural logarithm. This leverages the property \(\ln(e^x) = x\).
For the isolated equation \(\frac{5}{3} = e^{0.5 t}\), applying the natural logarithm \(\ln\) allows us to handle the equation in a logarithmic format: \(\ln(\frac{5}{3}) = 0.5 t\).
This step essentially "undoes" the exponential, simplifying our equation to a linear form. Finishing up involves solving for \(t\) by dividing both sides by 0.5: \(t = \frac{\ln(\frac{5}{3})}{0.5}\).
Thus, to solve such equations, knowledge of logarithmic properties is crucial. It allows us to transform and simplify equations for easier calculation and understanding.
For the isolated equation \(\frac{5}{3} = e^{0.5 t}\), applying the natural logarithm \(\ln\) allows us to handle the equation in a logarithmic format: \(\ln(\frac{5}{3}) = 0.5 t\).
This step essentially "undoes" the exponential, simplifying our equation to a linear form. Finishing up involves solving for \(t\) by dividing both sides by 0.5: \(t = \frac{\ln(\frac{5}{3})}{0.5}\).
Thus, to solve such equations, knowledge of logarithmic properties is crucial. It allows us to transform and simplify equations for easier calculation and understanding.
Other exercises in this chapter
Problem 10
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