Ch. 4 - Exponential and Logarithmic Functions

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Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problema 55

Capitolo 5, Problema 55

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. log₂(x+25)=4

Guida verificata passo dopo passo

Identify the given logarithmic equation: $\log_{2}(x + 25) = 4$.

Recall the definition of logarithm: $\log_{b}(A) = C$ means $b^{C} = A$. Apply this to rewrite the equation as $2^{4} = x + 25$.

Calculate \$2^{4}$ (without final numeric evaluation here) and set up the equation $x + 25 = 2^{4}$.

Solve for $x$ by isolating it: $x = 2^{4} - 25$.

Check the domain restriction for the logarithm: the argument $x + 25$ must be greater than 0, so ensure $x + 25 > 0$ and verify that your solution satisfies this condition.

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Properties of Logarithms

Understanding the basic properties of logarithms, such as the definition log_b(a) = c means b^c = a, is essential. This allows you to rewrite logarithmic equations in exponential form to solve for the variable.

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Change of Base Property

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) requires that the argument x be positive. When solving equations, it is crucial to check that solutions do not make the argument of any logarithm zero or negative, as these are not valid.

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Graphs of Logarithmic Functions

Exact and Approximate Solutions

After finding the exact solution to a logarithmic equation, it is often necessary to provide a decimal approximation. Using a calculator to round the solution to a specified number of decimal places helps interpret and communicate the result clearly.

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Graph Hyperbolas at the Origin

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