Ch. 4 - Exponential and Logarithmic Functions

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Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problema 76

Capitolo 5, Problema 76

In Exercises 74–79, solve each logarithmic equation. log2 (x+3) + log2 (x-3) =4

Guida verificata passo dopo passo

Step 1: Use the logarithmic property for addition, \( \log_b(A) + \log_b(B) = \log_b(A \cdot B) \), to combine the two logarithmic terms. The equation becomes \( \log_2((x+3)(x-3)) = 4 \).

Step 2: Simplify the expression \((x+3)(x-3)\) using the difference of squares formula, \( (a+b)(a-b) = a^2 - b^2 \). This gives \( \log_2(x^2 - 9) = 4 \).

Step 3: Rewrite the logarithmic equation in its exponential form. Recall that \( \log_b(A) = C \) implies \( b^C = A \). Here, \( 2^4 = x^2 - 9 \).

Step 4: Solve for \( x^2 \) by calculating \( 2^4 \), which equals 16, and then adding 9 to both sides of the equation. This gives \( x^2 = 16 + 9 \).

Step 5: Solve for \( x \) by taking the square root of both sides. Remember to include both the positive and negative roots, as \( x \) can be either \( \sqrt{25} \) or \( -\sqrt{25} \). Finally, check the solutions to ensure they do not make the original logarithmic expressions undefined.

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Logarithmic Properties

Understanding the properties of logarithms is essential for solving logarithmic equations. Key properties include the product rule, which states that log_b(m) + log_b(n) = log_b(m*n), and the power rule, which states that k*log_b(m) = log_b(m^k). These properties allow us to combine or simplify logarithmic expressions, making it easier to isolate the variable.

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Change of Base Property

Exponential Form

Logarithmic equations can often be solved by converting them into exponential form. For example, if log_b(a) = c, then a = b^c. This transformation is crucial for isolating the variable in the equation, as it allows us to express the logarithmic relationship in a more straightforward algebraic form.

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Exponential Functions

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. In the equation log2(x+3) + log2(x-3) = 4, both x+3 and x-3 must be greater than zero. This means that x must be greater than 3 for the logarithmic expressions to be defined, which is an important consideration when solving the equation.

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Graphs of Logarithmic Functions

Pratica correlata

Domanda del libro di testo

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. log(x+4)=log x+log 4

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Domanda del libro di testo

In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log_0.1 17

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Domanda del libro di testo

Find the domain of each logarithmic function. f(x) = log₅(x+4)

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Domanda del libro di testo

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