Questions: Complete parts a-d below. (a) Simplify log8(w^13) + log8(w^5). (b) Solve log8(w^13) + log8(w^5) = 1. (c) Compare the process of simplifying an expression with solving an equation. (d) Explain how simplifying an expression can help when you are solving an equation. (a) log8(w^13) + log8(w^5) = log8(w^18) (b) w ≈ 1.1225 (Type an integer or decimal rounded to four decimal places as needed.) (c) Which of the following is the correct comparison of simplifying an expression and solving an equation? A. Simplifying an expression gives a value that satisfies a given formula, while solving an equation is a process by which complicated logarithmic expressions are reduced. B. Simplifying an expression is a process by which complicated logarithmic expressions are reduced, while solving an equation gives a value that satisfies a given formula. C. Solving an equation is a way to find a value that simplifies an expression to a single logarithm. D. There is no difference. (d) Which of the following explains how simplifying an expression can help when you are solving an equation? A. Simplifying an expression given in an equation will make it easier to put the equation in logarithmic form and solve algebraically. B. Simplifying the expression in a given equation makes it easier to use the properties of logarithms. C. Simplifying an expression does not help when solving an equation. D. Simplifying an expression will help solve an equation because when the expression is fully simplified it can be changed from logarithmic form to exponential form, and then solved algebraically.

Complete parts a-d below. (a) Simplify log8(w^13) + log8(w^5). (b) Solve log8(w^13) + log8(w^5) = 1. (c) Compare the process of simplifying an expression with solving an equation. (d) Explain how simplifying an expression can help when you are solving an equation. (a) log8(w^13) + log8(w^5) = log8(w^18) (b) w ≈ 1.1225 (Type an integer or decimal rounded to four decimal places as needed.) (c) Which of the following is the correct comparison of simplifying an expression and solving an equation? A. Simplifying an expression gives a value that satisfies a given formula, while solving an equation is a process by which complicated logarithmic expressions are reduced. B. Simplifying an expression is a process by which complicated logarithmic expressions are reduced, while solving an equation gives a value that satisfies a given formula. C. Solving an equation is a way to find a value that simplifies an expression to a single logarithm. D. There is no difference. (d) Which of the following explains how simplifying an expression can help when you are solving an equation? A. Simplifying an expression given in an equation will make it easier to put the equation in logarithmic form and solve algebraically. B. Simplifying the expression in a given equation makes it easier to use the properties of logarithms. C. Simplifying an expression does not help when solving an equation. D. Simplifying an expression will help solve an equation because when the expression is fully simplified it can be changed from logarithmic form to exponential form, and then solved algebraically.
Transcript text: Complete parts a-d below. (a) Simplify $\log _{8}\left(w^{13}\right)+\log _{8}\left(w^{5}\right)$. (b) Solve $\log _{8}\left(w^{13}\right)+\log _{8}\left(w^{5}\right)=1$. (c) Compare the process of simplifying an expression with solving an equation. (d) Explain how simplifying an expression can help when you are solving an equation. (a) $\log _{8}\left(w^{13}\right)+\log _{8}\left(w^{5}\right)=\log _{8}\left(w^{18}\right)$ (b) $\mathrm{w} \approx 1.1225$ (Type an integer or decimal rounded to four decimal places as needed.) (c) Which of the following is the correct comparison of simplifying an expression and solving an equation? A. Simplifying an expression gives a value that satisfies a given formula, while solving an equation is a process by which complicated logarithmic expressions are reduced. B. Simplifying an expression is a process by which complicated logarithmic expressions are reduced, while solving an equation gives a value that satisfies a given formula. C. Solving an equation is a way to find a value that simplifies an expression to a single logarithm. D. There is no difference. (d) Which of the following explains how simplifying an expression can help when you are solving an equation? A. Simplifying an expression given in an equation will make it easier to put the equation in logarithmic form and solve algebraically. B. Simplifying the expression in a given equation makes it easier to use the properties of logarithms. C. Simplifying an expression does not help when solving an equation. D. Simplifying an expression will help solve an equation because when the expression is fully simplified it can be changed from logarithmic form to exponential form, and then solved algebraically.
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Solution

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Solution Steps

Solution Approach

(a) To simplify the expression \(\log_{8}(w^{13}) + \log_{8}(w^{5})\), use the logarithmic property that allows the sum of logs to be combined into a single log: \(\log_{b}(x) + \log_{b}(y) = \log_{b}(xy)\). This results in \(\log_{8}(w^{13} \cdot w^{5}) = \log_{8}(w^{18})\).

(b) To solve the equation \(\log_{8}(w^{18}) = 1\), convert the logarithmic equation to its exponential form: \(w^{18} = 8^{1}\). Then, solve for \(w\) by taking the 18th root of both sides.

(c) Simplifying an expression involves reducing it to a simpler form using mathematical properties, while solving an equation involves finding the value(s) of the variable(s) that satisfy the equation.

Step 1: Simplifying the Expression

To simplify the expression \(\log_{8}(w^{13}) + \log_{8}(w^{5})\), we use the property of logarithms that states \(\log_{b}(x) + \log_{b}(y) = \log_{b}(xy)\). Thus, we have:

\[ \log_{8}(w^{13}) + \log_{8}(w^{5}) = \log_{8}(w^{13} \cdot w^{5}) = \log_{8}(w^{18}) \]

Step 2: Solving the Equation

Next, we solve the equation \(\log_{8}(w^{18}) = 1\). By converting the logarithmic equation to its exponential form, we get:

\[ w^{18} = 8^{1} \]

This simplifies to:

\[ w^{18} = 8 \]

To find \(w\), we take the 18th root of both sides:

\[ w = 8^{\frac{1}{18}} \approx 1.1225 \]

Step 3: Comparing Simplifying and Solving

When comparing the processes of simplifying an expression and solving an equation, we find that simplifying an expression is a process by which complicated logarithmic expressions are reduced, while solving an equation gives a value that satisfies a given formula. Therefore, the correct comparison is:

\[ \text{B} \]

Final Answer

\[ \text{Simplified Expression: } \boxed{\log_{8}(w^{18})} \] \[ \text{Solution for } w: \boxed{1.1225} \] \[ \text{Comparison Answer: } \boxed{B} \]

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