Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problema 51

Capitolo 5, Problema 51

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. ln x=2

Guida verificata passo dopo passo

Recognize that the equation is given as $\ln x = 2$, where $\ln$ denotes the natural logarithm, which is the logarithm with base $e$ (Euler's number).

Rewrite the logarithmic equation in its equivalent exponential form. Recall that if $\ln x = 2$, then $x = e^2$.

Express the solution as $x = e^2$, which is the exact form of the answer.

Check the domain of the original logarithmic function. Since $\ln x$ is defined only for $x > 0$, verify that $e^2$ is positive, which it is, so no values are rejected.

If a decimal approximation is needed, use a calculator to evaluate $e^2$ and round the result to two decimal places.

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Properties of Logarithms

Understanding the natural logarithm function ln(x) is essential, where ln(x) is the logarithm base e. Key properties include ln(e) = 1 and the ability to rewrite logarithmic equations in exponential form, such as ln(x) = 2 becoming x = e^2.

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Change of Base Property

Domain of Logarithmic Functions

The domain of ln(x) is x > 0, meaning the argument inside the logarithm must be positive. When solving equations, any solution that results in a non-positive argument must be rejected to ensure the solution is valid within the function's domain.

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Graphs of Logarithmic Functions

Converting Between Logarithmic and Exponential Forms

Solving logarithmic equations often requires rewriting them in exponential form. For example, ln(x) = 2 can be rewritten as x = e^2, which allows direct computation of x. This conversion simplifies solving and interpreting logarithmic equations.

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Converting Standard Form to Vertex Form

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