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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 61
Capitolo 5, Problema 61

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 5 ln(2x)=20

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Start with the given equation: \(5 \ln(2x) = 20\).
Isolate the logarithmic expression by dividing both sides of the equation by 5: \(\ln(2x) = \frac{20}{5}\).
Simplify the right side: \(\ln(2x) = 4\).
Rewrite the logarithmic equation in its exponential form using the fact that \(\ln(a) = b\) means \(a = e^b\): \(2x = e^4\).
Solve for \(x\) by dividing both sides by 2: \(x = \frac{e^4}{2}\). Remember to check that this value of \(x\) keeps the argument of the logarithm positive, which it does since \(e^4 > 0\).

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Properties of Logarithms

Understanding the properties of logarithms, such as the ability to rewrite equations involving logarithms and exponents, is essential. For example, the natural logarithm ln(a^b) can be expressed as b ln(a), which helps in isolating variables and simplifying equations.
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Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function includes only positive arguments because the logarithm of zero or a negative number is undefined. When solving logarithmic equations, it is crucial to check that the solutions keep the argument inside the logarithm positive.
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Graphs of Logarithmic Functions

Solving Exponential Equations

After isolating the logarithmic expression, converting it to its equivalent exponential form allows solving for the variable. For example, if ln(y) = c, then y = e^c. This step is key to finding exact solutions before approximating decimal values.
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Solving Exponential Equations Using Logs
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